Is it permissible to seek treatment from a skilled Christian female doctor when less competent Muslim female doctors are available?

It is permissible for a Muslim to hire a non-Muslim for a specific task, just as the Prophet, peace and blessings be upon him, hired a polytheist guide during the Hijra.

The default is for a woman to go to a Muslim female doctor. If none is available, or if the non-Muslim doctor is more skilled, then there is no harm in going to her. The non-Muslim female doctor is preferred over a Muslim male doctor in the absence of a Muslim female doctor, because preserving life takes precedence.

Sheikh Ibn Uthaymeen permitted a Muslim woman to be treated by a trustworthy Christian woman, though it is preferable for the doctor to be Muslim.