The scholars are unanimously agreed that a Muslim who insults the Prophet (Sallallahu Alaihi Wasallam) becomes a Kaafir and an apostate who is to be executed.
Asma bint Marwan wrote a poem against Muhammad (Sallallahu Alaihi Wasalam) insulting him, Upon hearing the poem, Muhammad (Sallallahu alihi wa sallam) decided to have her executed.
Some scholars consider this story to be a forgery because the only sources for the story are Ibn Ishaq's Sira and Ibn Sa'd's Kitab al-Tabaqat al-Kabir. Also, one of the reporters in the chain of narration, Muhammad Ibn Al-Hajjaj, is known for fabricating Hadiths.
Ibn Sa'd's Kitab al-Tabaqat al-Kabir describes the death of Asma bint Marwan:
"Then (occurred) the Sariyyah of Umayr ibn adi Ibn Kharashah al-Khatmi against Asma Bint Marwan, of Banu Umayyah Ibn Zayd, when five nights had remained from the month of Ramadan, in the beginning of the nineteenth month from the Hijrah of the apostle of Allah. Asma was the wife of Yazid Ibn Zayd Ibn Hisn al-Khatmi. She used to revile Islam, offend the Prophet and instigate the (people) against him. She composed verses. Umayr Ibn Adi came to her in the night and entered her house. Her children were sleeping around her. There was one whom she was suckling. He searched her with his hand because he was blind, and separated the child from her. He thrust his sword in her chest till it pierced up to her back. Then he offered the morning prayers with the Prophet at al-Medina. The apostle of Allah said to him: "Have you slain the daughter of Marwan?" He said: "Yes. Is there something more for me to do?" He [Muhammad] said: "No. Two goats will butt together about her. This was the word that was first heard from the apostle of Allah. The apostle of Allah called him Umayr, "Basir" (the seeing).
(Ibn Sa'd' Kitab al-Tabaqat al-Kabir, translated by S. Moinul Haq, volume 2, page 31 )
Ibn 'Adiyy mentions this incident in his book "Al-Kamal" on the authority of Ja'far Ibn Ahmad Ibn Muhammad Ibn As-Sabah on authority of Muhammad Ibn Ibrahim Ash-Shami on authority of Muhammad Ibn Al-Hajjaj Al-Lakhmi on authority of Mujalid on authority of Ash-Shu'abi on authority of Ibn 'Abbas, and added that
"...this Isnad (chain of reporters) is not narrated on authority of Mujalid but by Muhammad Ibn Al-Hajjaj and they all (other reporters in the chain) accuse Muhammad Ibn Al-Hajjaj of forging it." (Ibn 'Adiyy, Al-Kamal, Vol. 6, p. 145)
In the authentic Sunnah of the Prophet (Sallallahu Alaihi Wa sallam) prohibits the killing of women in war. (note: in Islam a women can only be put to death if she commited murder as crime)
Narrated Ibn 'Umar: Messenger of Allah (Sallallahu Alaihi Wasallam) saw the corpse of a woman who had been slain in one of the raids, and he disapproved of it and forbade the killing of women and children. (Bukhari, Vol. 4, Bk. 52, No. 257 & 258).
Due to this prohibition, scholars of Abu Hanifah’s Madhab (school of thought) have stated that apostate women are not to be killed because the Prophet (Sallallahu Alaihi Wasallam) forbade the killing of women, and since the prohibition is general it includes apostate women. (Sharh Ad-Durr-el-Mukhtar, Volume 1, p. 483, 'Ainul Hidayah commentary of Al Hidayah Vol. 2 Pg. 585)
Even after the Prophet's demise, his Sunnah remain preserved by the Muslims:
Abu Bakr advised Yazid: "I advise you ten things: Do not kill women or children or an aged, infirm person. Do not cut down fruit-bearing trees. Do not destroy an inhabited place. Do not slaughter sheep or camel except for food. Do not burn bees and do not scatter them. Do not steal from the booty, and do not be cowardly." (Mutta Malik, Book 21, Section 3, Number 10)
Therefore, based on the evidences, we can therefore conclude that the killing of Asma' bint Marwan are false.