Is it permissible for my mother to perform tayammum and pray, even though the prayer time has passed by a short while, due to an illness in her hand?

If the mother is certain that using water for ablution (Wudu') will aggravate her illness, delay her recovery, or cause a new illness, then it is permissible for her to perform Tayammum (dry ablution). As for delaying the prayer, if it does not take it out of its preferred time, then there is no harm. However, if she delays it until the preferred time expires and the necessary time begins without a legitimate excuse, then she is sinful, and her prayer is considered an اداء (performance within its due time). As for delaying it until the necessary time expires, this is not permissible except for an excuse such as sleep or forgetfulness; otherwise, she is disobedient, and her prayer is considered a قضاء (make-up).