الجواب وبالله التوفيق

Your mother is right, lactation stops due to fasting and the baby becomes extremely upset due to hunger, which can be damaging to the child. That is the reason why nursing mothers are allowed to skip fasting during the month of Ramadhan but are required to make up for the missing fasts afterwards. It is impermissible to pay fidyain this situation and whatever has been paid as fidya will be considered as sadaqah. During the rest of her life she can make up for these missing fasts at any point.

(عن انس بن مالک الکعبی قال قال رسول اللہ صلی اللہ علیہ وسلم ان اللہ وضع عن المسافر شطر الصلوۃ والصوم عن المسافر وعن المرضع والحبلٰی(رواہ الداؤد والترمذی والنسائی وابن ماجہ و مشکٰوۃ

لمن خاف زیادۃ المرض الفطر و للمسافر ۔۔۔وللحامل والمرضع ان خافتا علی الولد اوالنفس

 (حاشیہ کفایت المفتی ص301، ج6، بحوالہ بحرالرائق کتاب الصوم)

واللہ اعلم بالصواب