Is Hebrews 10:26 mistranslated?

I appreciate the OP's effort to understand this tricky verse that is Heb 10:26. Before giving my own translation, let me offer some comments on the proposed OP's translation.

• there is no third person pronoun in the Greek equivalent to "they". The closest we get is a first person genitive pronoun ἡμῶν = "of us".
• there is no "if" in the Greek either. Such is an interpretation by both the OP and most versions. However, such is certainly implied in the Greek construction.
• the verb "believe" is also absent from the Greek and thus is a theological interpretation by the OP.
• the word "continually" is also absent in the Greek
• "don't" is also absent in the Greek
• The pronoun ἡμῶν is genitive and thus must be translated either "our" or "of us" but not "from us" - the latter would be closer to the dative case.

OK, so here is my attempt to very literally render the Greek into sensible (but rather awkward) English:

For [if] our sinning willingly after the receiving the knowledge of the truth, no longer for of sins a sacrifice remains.

The above does not contain typos and is somewhat deliberately awkward to show the original Greek. Most modern versions will then smooth out the English to something like:

If we deliberately go on sinning after we have received the knowledge of the truth, no further sacrifice for sins remains,

Such a teaching is quite consistent with the immediate context:

• Heb 6:4-6 - It is impossible for those who have once been enlightened, who have tasted the heavenly gift, who have shared in the Holy Spirit, who have tasted the goodness of the word of God and the powers of the coming age— and then have fallen away—to be restored to repentance, because they themselves are crucifying the Son of God all over again and subjecting Him to open shame.
• Heb 10:29 - How much more severely do you think one deserves to be punished who has trampled on the Son of God, profaned the blood of the covenant that sanctified him, and insulted the Spirit of grace?
• Heb 10:35, 36 - So do not throw away your confidence; it holds a great reward. You need to persevere, so that after you have done the will of God, you will receive what He has promised.

That is, the author of Hebrews is teaching that it is possible to lose one's faith and be lost.

No. It is absolutely critical to read the scriptures from the first audience perspective. The book of Hebrews was written to the believers in Christ about 65-66 AD. (DatingTheNewTestament). There is no need to try to change the words, only to change our view point.

How shall we escape, if we neglect so great salvation; which at the first began to be spoken by the Lord, and was confirmed unto us by them that heard him; (Heb. 2:3, YLT)

The word "hebrew" was assigned to Abraham before there was a tribe of Israel, and it meant a foreigner, one who had crossed over from the other side of the river Frat. From Barne's Notes at Gen. 14:13 -

“At all events, this is the first appearance of the name in Scripture. As we all know, it has become that of the nation, but a Jew did not call himself a ‘Hebrew’ except in intercourse with foreigners. As in many other cases, the national name used by other nations was not that by which the people called themselves. Here, obviously, it is not a national name, for the very good reason that there was no nation then. It is a personal epithet, or, in plain English, a nickname, and it means, probably, as the ancient Greek translation of Genesis gives it, neither more nor less than ‘The man from the other side,’ the man that had come across the water.” (BibleHub)

Excerpt from my post Crossing Over:

“Êber” in the Hebrew means “on the other side of.” The ancestors of Israel are described as those who “dwelt of old time beyond the River” (êber ha-nâhâr = “on the other side of the Euphrates river”).”

Therefore Eber is presumed to have crossed over either the Tigris or the Euphrates. Then his descendant Abram crossed over the Euphrates, also called the river Frat.

Jacob crossed the rivers Euphrates and Jordan both when leaving his father Isaac’s land in Canaan, and again upon his return from Laban in Haran (Gen 31:21; 32:10). It was at the brook, or stream of Jabbok, which flows to the Jordan where Jacob wrestled with the angel (Gen. 32:24-31; 48:15-16), and received the blessing and a new name “Israel” as a prince of God. And, after having been blessed, in the morning he crossed over the brook of Jabbok.

Is it long before we think of Abram’s descendants crossing over the Red Sea in their exodus from Egypt? Then in order to enter the physical promised land of Canaan, they again cross over water, the river Jordan.

3 And I took your father Abraham from the other side of the flood, and led him throughout all the land of Canaan, and multiplied his seed, and gave him Isaac. (Josh. 24:3, KJV)

God called Abram out of Haran. He called Abraham’s descendants through Isaac and Jacob (Israel) out of Egypt. Each had to cross over the water to the land God provided.

Elijah crossed the river Jordan on dry land before he was taken up in the chariot (2 Kings 2:4-8). Elisha, when he took up Elijah’s mantle, crossed back over the river Jordan on dry ground before beginning his ministry for YHVH (2 Kings 2:13-14).

Jonah was immersed in water in the belly of the fish, with the surging waves passing over him before beginning his ministry to Nineveh (Jon. ch. 2).

This crossing through, passing through the waters, is an anointing of God: type and anti-type.

God now calls us through His Son, Yeshua (Jesus) to come out of the darkness and be separate for Him (1 Pet. 2:9), to be His priests and kings on this earth (Rev. 5:10). A symbolic part of that calling out is crossing the water, which anti-type is now immersion.

And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord. (Acts 22:16, KJV) ....

16 As soon as Yeshua had been immersed, he came up out of the water. At that moment heaven was opened, he saw the Spirit of God coming down upon him like a dove, 17 and a voice from heaven said, “This is my Son, whom I love; I am well pleased with him.” (Matt. 3:16-17, CJB)

Jesus was anointed by the Holy Spirit after he crossed the water, entered into it, immersed himself in it, and rose back up. He crossed the same water, the river Jordan, as had the Israelites whom God had called out of bondage in Egypt (Josh 3:14-16; John 1:28). Jesus’ crossing of the river Jordan was at that very same spot where the Israelites had entered into the Jordan. (9) (10) It was the anti-type and fulfillment of the OT crossing.

This is being born of the water and the spirit (John 3:5). We cross the water (immersion/baptism) from darkness into light, from bondage and slavery into freedom from sin and death; and then the Holy Spirit will anoint us, and write our names in the book of life (Phil. 4:3, Rev. 21:27). Only then are we in Christ (Rom. 6:3; Gal. 3:27). ....

4 Yes, indeed! I tell you that whoever hears what I am saying and trusts the One who sent me has eternal life — that is, he will not come up for judgment but has already crossed over from death to life! (John 5:24, CJB) ...

And, now we ask…why is the book of Hebrews called “Hebrews”? Isn’t it addressed to those who have already crossed over?

So, from this analysis we conclude the book was written to Christians who had already been baptized. Therefore, we must look at the meaning of Heb. 10:26 from the first audience perspective of those first century AD Christians who were living while the animal sacrifices were still on-going under the Mosaic Law at the temple in Jerusalem before it was destroyed by the Romans.

We need to keep this picture in our minds. Every sin we commit is a willful sin because we chose our actions, or we chose not to act when we should. The point is not on the willfulness of sinning, but on the sacrifice needed for repentance of the sins. The scripture is saying that there is no need to keep taking animal sacrifices to the temple to take our place to be killed. Jesus already did that for us. There is NO OTHER sacrifice required.

Excerpt from my post No More Sacrifice For Sin -

...In AD 65-66 the writer of Hebrews was explaining how this old system was about to pass away (Heb 8:13), and that the old system under the Mosaic Law was no longer effective as bulls and goats could not take away sin (Heb. 10:4), and yet the sin offerings were still happening at the temple in Jerusalem. Picture this background in your mind.

The point of Heb. 10:26 was that those who had been immersed into Christ, who had been immersed into His sacrifice (Rom. 6:3-6),had no need to make any other offering. The words, “there no longer remains a sacrifice for sin,” at Heb. 10:26 means that no more animal sacrifices will work. They didn’t need to keep taking animals to the temple any more. God was telling them that system was done and over. ...

A better way for us to understand Heb. 10:26 in our day and time might be to say “there remains no OTHER sacrifice for sins”. If we make a mistake and willfully sin after having been immersed into Christ, we can still repent. Our only sacrifice is Christ Jesus! No other sacrifice will be acceptable to God for forgiveness to the truly repentant heart.

If we continue to repeatedly sin willfully, then we have scorned Christ, trampled upon His sacrifice, and are not repentant. We will not be forgiven if we do not repent and stop doing those sins.

Of how much sorer punishment, suppose ye, shall he be thought worthy, who hath trodden under foot the Son of God, and hath counted the blood of the covenant, wherewith he was sanctified, an unholy thing, and hath done despite unto the Spirit of grace?” (Heb. 10:29, YLT)

But having accepted Christ, having been immersed into Christ, if we sin again, we do not have to sacrifice a dove, or a lamb, or a bull to be forgiven. We only have to turn back to Christ, and repent, and stop sinning. It’s the old Bob Newhart joke of a psychiatrist who tells his patient that is complaining of getting hurt each time she does something, and his says he has two words for her: “STOP IT!.”

Heb 10:26 is correctly understood as written when we know how the first audience perceived it.

I quote two renown New Testament Greek grammarians. Ἑκουσίως ... ἁμαρτανόντων ἡμῶν NA28 (a genitive absolute clause, "if we go on sinning deliberately" ESV); μετὰ τὸ λαβεῖν τὴν ἐπίγνωσιν τῆς ἀληθείας,... NA28 (Daniel Wallace used this as an example an adverbial clause of time "after ..." "after receiving the knowledge of the truth" ESV). If the grammar of this verse allowed a translation eased the meaning, A.T. Robertson would have pointed it out. I would expect the same from Daniel Wallace.

A Greek writer wouldn't have made a genitive absolute clause as ambiguous as your experimental translation.

Craig Keener interprets Heb. 10:26 in light of the sacrificial laws and applies this verse to unrepentant sin.

Grammars

C. Time

1. Antecedent (μετὰ τό + infinitive) [after …]

The action of the infinitive of antecedent time occurs before the action of the controlling verb. Its structure is μετὰ τό + the infinitive and should be translated after plus an appropriate finite verb....

Heb 10:26 ἑκουσίως ἁμαρτανόντων ἡμῶν μετὰ τὸ λαβεῖν τὴν ἐπίγνωσιν τῆς ἀληθείας if we sin deliberately after receiving the knowledge of the truth -- Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 594-595). Zondervan.

Hebrews 10:26

If we sin wilfully (ἑκουσιως ἁμαρτανοντων ἡμων [hekousiōs hamartanontōn hēmōn]). Genitive absolute with the present active participle of ἁμαρτανω [hamartanō], circumstantial participle here in a conditional sense. After that we have received (μετα το λαβε��ν [meta to labein]). “After the receiving” (accusative case of the articular infinitive second aorist active of λαμβανω [lambanō] after μετα [meta]). Knowledge (ἐπιγνωσιν [epignōsin]). “Full knowledge,” as in 6:4f. There remaineth no more (οὐκετι ἀπολειπεται [ouketi apoleipetai]). “No longer is there left behind” (present passive indicative as in 4:9), for one has renounced the one and only sacrifice for sin that does or can remove sin (10:1–18). -- Robertson, A. T. (1933). Word Pictures in the New Testament (Heb 10:26). Broadman Press.

Commentary

10:26. Judaism had long distinguished intentional and unintentional sin (Num 15:29–31; cf., e.g., Lev 4:2, 22); one who knew better would be punished more strictly than one who was ignorant. Sacrifices *atoned for sins of ignorance, but Judaism taught that no sacrifice availed for the person who knowingly rejected the authority of God’s *law. (For such persons, many Jewish teachers insisted that *repentance, the Day of Atonement and death were all necessary. Jewish teachers also observed that those who sinned presuming that they would be automatically forgiven were not genuinely repentant and hence were not forgiven.) In the *Dead Sea Scrolls, slight transgressions required temporary penance, but deliberate rebellion against God’s law demanded expulsion from the community. The sin in this context is unrepentant, thorough apostasy (10:29). -- Keener, C. S. (2014). The IVP Bible Background Commentary: New Testament (Second Edition, p. 656). IVP Academic: An Imprint of InterVarsity Press.

I am not an expert on Greek. Hebrews is written to people who had been living under the law of Moses before believing in Jesus. By the sound of Hebrews 10:26 and Hebrews 6:4-8, it was possible for those first Israelite Christians to believe in Jesus and then go back to thinking animal sacrifices in the temple could get their forgiveness of sins, and so unbelieve in Jesus, and then lose their salvation. I think it was only valid up until 70 AD when the temple was destroyed, and it is no longer possible.

Another reference to judgment, fire on Christians is here. It is not loss of salvation fire, though like the Hebrews verses imply.

1 Corinthians 3:10-15

¹⁰ By the grace God has given me, I laid a foundation as a wise builder, and someone else is building on it. But each one should build with care. ¹¹ For no one can lay any foundation other than the one already laid, which is Jesus Christ. ¹² If anyone builds on this foundation using gold, silver, costly stones, wood, hay or straw, ¹³ their work will be shown for what it is, because the Day will bring it to light. It will be revealed with fire, and the fire will test the quality of each person’s work. ¹⁴ If what has been built survives, the builder will receive a reward. ¹⁵ If it is burned up, the builder will suffer loss but yet will be saved—even though only as one escaping through the flames.

Your interpretation of the Greek is just wrong.

The words ἁμαρτανόντων ἡμῶν are a construction called a genitive absolute. The two words go together, agreeing in number, gender, and case. A genitive absolute is an add-on to a sentence that describes the circumstances under which the main action of the sentence happened. For example, "The weather being rainy, we canceled the picnic." These two words in Hebrews mean "we giving witness," so the sentence says "For willingly, we having borne witness, ..."

The word οὐκέτι simply means "no more, no longer, not in the future." It always implies a statement about time. Here is the entry from a standard dictionary: https://lsj.gr/wiki/%CE%BF%E1%BD%90%CE%BA%CE%AD%CF%84%CE%B9