Is it permissible for us to hit a brother who is not of sound mind when he harms us, in order to silence him?

You must be patient with your brother and not harm him by striking him. Instead, use gentle and benevolent methods to draw him closer, for kindness adorns everything, and its removal disgraces it. It is likely that beating a madman will not benefit him, though if it serves to discipline and is permissible, it must be light and not severe. Otherwise, it is not permissible. As for mocking a madman and ridiculing him, this could lead to the loss of one's own sanity as a punishment. Therefore, one should fear Allah and praise Him for good health, and ask Allah to grant the madman health, for madness is a trial from Allah and not by his choice.