I am using ENC28J60 and EthernetENC library to try to connect Arduino UNO with ATmega328 to a network. It works fine within the local network, but outside of it, the ping does not go through. The network settings are 100% correct (other devices are working normally). The AdvancedChatServer example from the Ethernet library was able to be pinged for a while, but after some time, it stopped responding to pings. Other examples do not respond to pings at all.

////////////////////////////////////// Settings //////////////////////////////////////
#define DHCP false          // true/false
#define CUSTOM_MAC true     // true/false

#if CUSTOM_MAC              // if true you can enter your custom mac address
#define MAC 0x00,0x5f,0x1C,0x05,0x5f,0x1C
#else
#define MAC 0x00,0x5f,0x1C  // Organizationally Unique Identifier (OUI)
#endif

#define IP_ADDRESS  192,168, 11,242
#define SUBNET      255,255,255,248
#define GATEWAY     192,168, 11,241
#define DNS         192,168, 11,241


// #define UIP_CONF_MAX_CONNECTIONS 0
///////////////////////////////////// Libraries //////////////////////////////////////
#include <Arduino.h>
#include <EthernetENC.h>  // Include the Ethernet library
#include <Agentuino.h>    // SNMP library
#include <EEPROM.h>
#include <microDS18B20.h>
/////////////////////////////// Arduino sensor pins //////////////////////////////////
#define TEMPERATURE_1_PIN 0
#define TEMPERATURE_2_PIN 1
#define TEMPERATURE_3_PIN 2
#define TEMPERATURE_4_PIN 3
#define SENSOR_220V_PIN   4
#define SERVO_PIN         5
#define LED_ALERT_PIN     6
#if !SOUND_MUTE
#define SPEAKER_PIN       7
#else
#define SPEAKER_PIN       8
#endif
#define FLOOD_PIN         A0
#define DOOR_PIN          A1
#define GAS_PIN           A2


///////////////////////////// Ethernet / SNMP init ///////////////////////////////////
#if !DHCP
IPAddress ip(IP_ADDRESS);   // IP Address for arduino
IPAddress gateway(GATEWAY); // Gateway for arduino
IPAddress subnet(SUBNET);   // SubNet mask for arduino
IPAddress myDns(DNS);       // DNS for arduino
#endif

byte mac[] = {MAC};

//////////////////////////////////// My OIDs /////////////////////////////////////////
...
//////////////////////////////////////////////////////////////////////////////////////

/////////////////////////////// function SNMP resived ////////////////////////////////
void pduReceived(){...}
//////////////////////////////////////////////////////////////////////////////////////

void setup(){
  ///////////////////////////////// init pins ////////////////////////////////////
  pinMode(TEMPERATURE_1_PIN, INPUT);
  pinMode(TEMPERATURE_2_PIN, INPUT);
  pinMode(TEMPERATURE_3_PIN, INPUT);
  pinMode(TEMPERATURE_4_PIN, INPUT);
  pinMode(SENSOR_220V_PIN, INPUT);
  // pinMode(SERVO_PIN, INPUT);
  pinMode(FLOOD_PIN, INPUT);
  pinMode(DOOR_PIN, INPUT);
  pinMode(GAS_PIN, INPUT);
  pinMode(LED_ALERT_PIN, OUTPUT);
  #if !SOUND_MUTE
  pinMode(SPEAKER_PIN, OUTPUT);
  #endif

  ////////////////////////////// init other //////////////////////////////////////
  Serial.begin(9600); // Serial monitor init
  Agentuino.begin();  // Begin Snmp agent on Ethernet shield

  ///////////////////////// Generate random mac address //////////////////////////
#if !CUSTOM_MAC
  if (EEPROM[0] == 255 && EEPROM[1] == 255 && EEPROM[2] == 255){
    for (int i = 3; i < 6; i++){
      randomSeed(analogRead(A5)); 
      mac[i] = random(0xFF);
      EEPROM.put(i - 3, mac[i]);
    }
  }
  else{
    for (int i = 3; i < 6; i++){ 
      EEPROM.get(i - 3, mac[i]); // Reed saved mac addres from EEPROM
    }
  }
#endif

#if DHCP
  Ethernet.begin(mac) // Initialize Ethernet Shield in DHCP mode
#else
  Ethernet.begin(mac, ip, myDns ,gateway, subnet); // Initialize Ethernet Shield in static mode
#endif

///////////////////////////////////// Debug //////////////////////////////////////
Serial.println();
Serial.println();
  if (Ethernet.hardwareStatus() == EthernetNoHardware) {
    Serial.println(F("Ethernet shield was not found.  Sorry, can't run without hardware. :("));
    while (true) {
      delay(1); // do nothing, no point running without Ethernet hardware
    }
  }
  if (Ethernet.linkStatus() == LinkOFF) {
    delay(500);
    if (Ethernet.linkStatus() == LinkOFF) {
      Serial.println("Ethernet cable is not connected.");
    }
  }

  Serial.print(F("localIP: "));
  Serial.println(Ethernet.localIP());
  Serial.print(F("dnsServerIP: "));
  Serial.println(Ethernet.dnsServerIP());
  Serial.print(F("gatewayIP: "));
  Serial.println(Ethernet.gatewayIP());
  Serial.print(F("subnetMask: "));
  Serial.println(Ethernet.subnetMask());
  Serial.print(F("mac: "));
  for (int i = 0; i < 6; i++){
    Serial.print(mac[i], HEX);
    if (i <= 4) Serial.print(F("."));
  }
  
  Serial.println();
  ////////////////////////////////////////////////////////////////////////////////
  
  Agentuino.onPduReceive(pduReceived);
  delay(10);
}


void loop(){
  Agentuino.listen();

  //////////////////////////////// Other code ////////////////////////////////////
  ...
  ////////////////////////////////////////////////////////////////////////////////
}