Al-Haafith ibn Hajar (rahimahullaah) wrote the biography of this female companion in his book Al-Isaabah, and we will quote part of it with slight modifications. He wrote,
"Her name is Asmaa' bint 'Umays ibn Ma'd from the tribe of Khath'am. She was the maternal sister of Maymoonah bint Al-Haarith the wife of the Prophet (peace be upon him), and sister to a group of female Companions. She was one of the early female Companions who migrated to Abyssinia as she migrated there with her husband Ja'far ibn Abi Taalib (may Allaah be pleased with him). She gave birth to his children there. When Ja'far was killed, Abu Bakr (may Allaah be pleased with him) married her, and she gave birth to his son Muhammad. Later she got married to 'Ali (may Allaah be pleased with him) and it is said that she gave birth to his son 'Awn."
Ibn Hajar did not mention the date of her death, but in "Taqreeb At-Tahtheeb", he said that she died after 'Ali and he did not mention the year in which she died. We did not find any scholar mentioning the year of her death.