Is it permissible to pray while an infant is sitting on the prayer rug?

A young child sitting on a prayer rug does not invalidate it, unless it leads to an impurity that affects the worshiper or their place. Minor actions done out of necessity, such as carrying or moving the child, do not cause harm. Ibn Qudamah stated that this is permissible, citing the hadith of Abu Qatadah that the Prophet, peace and blessings be upon him, carried Umamah bint Zaynab during his prayer.