Is hitting the buttocks, which is a private area, permissible Islamically as a disciplinary punishment for students?

We advise against hitting children as much as possible, and that disciplining them with gentleness and kindness is better, for the Prophet, peace and blessings be upon him, never hit a servant or a woman. Hitting for discipline is permissible when necessary, such as for commanding prayer after the age of ten, and it should be a light, non-injurious hit. We do not know what prevents hitting the buttocks if there is no fear of harm or if it is not injurious, while avoiding hitting the face and anything that might cause death or severe harm such as the stomach and genitals.