Is my mother still Muslim after she jokingly said, "I am an infidel"? And what are the implications for her marriage if that is the case?

It is not permissible to provoke the mother and repeatedly question her about disbelief (kufr), because this action by the son is a form of disobedience to parents (uquq al-walidayn), from which he must repent. If the mother willingly admits to disbelief, she becomes a disbeliever by that, even if she was joking. If she then pronounces the two testimonies (Shahadatayn) or prays, she is considered a Muslim. The son must be careful not to joke with his mother in such a manner, and he should gently explain to her the seriousness of admitting to disbelief.