Is it permissible to hit one's wife, given the prohibition of hitting animals, and is it true that one of the prophets hit his wife, and if so, why?

Hitting a wife for reasons other than nushuz (disobedience/rebellion) is forbidden. Even with nushuz, severe beating is forbidden. Islam permitted light striking that leaves no mark. Islam rejects the doubts propagated by its enemies regarding the status of women. It commands the husband to provide proper housing, clothing, and food for his wife; otherwise, she has the right to seek divorce, and the ruler obliges the husband to provide maintenance or divorce, and he may be imprisoned if he refuses. Meanwhile, disbelievers have taken women out of their homes, made them cheap commodities, and burdened them with arduous labor. Allah Almighty spoke truthfully: "Neither those who disbelieve from the People of the Scripture nor the polytheists wish that any good should be sent down upon you from your Lord," and "They wish that you would disbelieve as they disbelieved, so you would be alike."