It's definitely a thing in modern Arabic literature, at least since the second half of the 20th century.
Thanks to Malady's answer providing the Arabic term for a very short story, "al-qissa al-qasira jiddan", I was able to find the following article, on which this answer is based: Ibrahim Taha, "The Modern Arabic Very Short Story: A Generic Approach", Journal of Arabic Literature 31(1) (2000), pp. 59-84.
On the origins of the very short story in Arabic literature:
As for the very short story in modern Arabic literature, from the viewpoint of the sociology of literature a certain connection exists between this form and the scientific development that the world has experienced in recent decades, with its effect on all aspects of modern life. Numerous revolutionary, technological, and cultural information sources have contributed to the appearance of very short fiction texts compatible with the fast pace of this development. Edward al-Kharrat observes that additional reasons for the creation of this genre relate to political conditions and the serious consequences of the 1967 war for the Arab world - harsh social conditions and disappointment with socialist ideals. From a literary point of view, al-Kharrat adds the major factor of the failure of the short story, as an institutionalized genre, to constantly innovate and explore new possibilities.2 Generally, the state of openness, fluidity, and blurring of borders among information, culture, and art appears to have contributed greatly to the interaction among various genres. In our case, the blurring of borders between the short story and poetry has led to the birth of the very short story, as will be discussed later on.3
(Links added by me for context. Footnotes refer to Edward al-Kharrat, al-Kitaba 'abr al Naw'iyya: Maqalat fi Zahirat al-Qissa al-Qasida wa-Nusus Mukhtara, Cairo, Dar Sharqiyyat, 1994, p. 20, and Khayri Duma, Tadakhul al-Anwa' fi al-Qissa al-Misriyya al-Qasira 1960-1990, Cairo, al-Hay'a al-Misriyya al-'Amma lil-Kitab, 1998 [in footnote 2] and Ahmad 'Abd al-Raziq Abu al-'Ula, "al-Qissa al-Qasira Jiddan wa-Ishkaliyyat al-Shakl al-Qasasi al-Mu'asir," al-Thaqafa al-Jadida, August (1989), pp. 40-43; Ghali Shukri, "Min Ishkaliyyat al-Qissa al-Misriyya al-Qasira," Ibda', August (1989), pp. 7-19 [in footnote 3].)
On the volume of very short stories existing in 20th-century Arabic literature:
The number of Arabic texts written under the generic auspices of the very short story is extremely large and is constantly increasing.4
4 This genre has increasingly been the writing focus of many Arab writers, for example, Yahya al-Tahir 'Abdallah, Hana' 'Atiyya, 'Abd al-Mun'im al-Baz, Rifqi Badawi, Badr al-Dib, Nadir al-Halawani, Sa'd al-Din Hasan, 'Abd al-Hakim Haydar, Muhammad al-Makhzanji, Muahammad Mustajab, Nabil Na'um, Muntasir al-Qaffash, 'Ibtihal Salim, Sayyid al-Wakil (Egypt); Mahmud al-Rajabi (Jordan); Salim al-Humaydi (Yemen); Mahmud Shuqayr (Palestine); Zakariyya Tamir (Syria); Ibrahim Ahmad, Musin al-Khafaji (Iraq), Muhammad Nujaym (Morocco), and many others.
This gives us a nice list of authors who have been active in writing very short stories, and perhaps there will be more questions here on Literature SE about some of their works ...
