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.