Is it permissible for a doctor to perform a tubal ligation on a woman who is about to give birth for the fourth time via C-section, out of fear for her future demise?

It is advisable to consult doctors to determine the level of danger. It is absolutely forbidden to perform a sterilization operation (vasectomy/tubal ligation) because it undermines the objective of Sharia in increasing progeny. However, if a trustworthy Muslim doctor confirms medically that pregnancy poses a danger to the woman's life or significant harm, then there is no objection to performing a procedure to prevent pregnancy, provided it is performed by a woman, not a man.