Is it permissible for a girl to vow herself to the Messenger of Allah, may Allah bless him and grant him peace, and not marry?

This vow is not a vow of devotion and does not necessitate fulfillment. It is a nullity, and there is no expiation required for the one who made such a vow. The outcome of this vow leads to celibacy and abstaining from marriage in this world, which is not legislated; rather, it contravenes the Sunnah of the Prophet, peace and blessings be upon him.