All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
‘Asmaa' bint Marwaan was from (the tribe of) Banu Umayyah ibn Zayd. She was the wife of Yazeed ibn Zayd ibn Hisn Al-Khatmi Al-Ansaari.
As for the story of her killing, it was cited in the biographical books (Siyar), but many scholars declared it inauthentic. In fact, a number of researching scholars, such as Ibn ‘Adiyy, Ibn Al-Jawzi, and Al-Albaani, said that it was false and fabricated and that Muhammad ibn Al-Hajjaaj, who was declared a liar, was accused of fabricating it.
In brief, the story, according to Maghaazi Al-Waaqidi and Seerat Ibn Hishaam is that she used to write poetry defaming Islam; offending the Prophet, sallallaahu ‘alayhi wa sallam; instigating people against him; and throwing filth (rags smeared with menstrual blood) in the mosque of her people, Banu Khatmah. The Prophet, sallallaahu ‘alayhi wa sallam, declared that her blood was made lawful on account of her hideous actions. A Companion named ‘Umayr ibn ‘Addiyy from her own tribe killed her in the second Hijri year, after the Battle of Badr.
You can read her detailed story in the books on biographies. However, we would like to draw your attention to the fact that such books contain authentic, inauthentic, and fabricated reports. Therefore, Al-‘Iraaqi said, "The reader should know that the 'Siyar' compile what is authentic and what is rejected." This story, in particular, was classified as fabricated and false by some scholars.
Allah knows best.