Is it permissible to divorce a woman who is breastfeeding?

Yes, it is permissible to divorce a woman who is breastfeeding, if the husband divorces her during a period of purity in which he has not had intercourse with her. But if he divorces her when she is menstruating, or during a period of purity in which he has had intercourse with her, then this is a prohibited divorce according to scholarly consensus. 

Ibn Qudaamah (may Allah have mercy on him) said: 

As for that which is prohibited, it is divorce during menses, or during the period of purity in which he has had intercourse with her. 

The scholars of all regions and all time periods are unanimously agreed that that is prohibited, and is called an innovated divorce (talaaq al-bid‘ah), because the one who issued this divorce has gone against the Sunnah and ignored the command of Allah, may He be exalted, and His Messenger. Allah, may He be exalted, says (interpretation of the meaning): “divorce them at a time when the period of ‘Iddah may start” [at-Talaaq 65:1]. And the Prophet (blessings and peace of Allah be upon him) said: “If he (the husband) wants to issue a divorce before he has touched his wife, he may do so; that is the time at which Allah has permitted divorcing women.” End quote. 

Al-Mughni (8/235) 

If the breastfeeding woman is not menstruating because of breastfeeding, as often happens, then if her husband has not had intercourse with her during that period of purity, he may divorce her, but if he has had intercourse with her, then it is not permissible for him to divorce her until she menstruates, then becomes pure, even if that takes a long time. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

The words: “During a period of purity in which he has not had intercourse with her” exclude the case in which he divorces her during a period of purity in which he has had intercourse with her, because that is an innovated divorce, even if the period of purity lasted for a long time. If we assume that this man’s wife became pure from nifaas (post-partum bleeding) and he had intercourse with her when she was still breastfeeding, and usually the woman who is breastfeeding does not menstruate until she has weaned her child, which is after approximately two years, then if he divorces her within those two years, it is an innovated divorce, because it has occurred during a period of purity in which he had intercourse with her. So he should wait until her menses has come then she becomes pure. End quote.

Ash-Sharh al-Mumti‘ (13/37)

And Allah knows best.