Do you mean a British passport? Which name would you prefer to use?
There is nothing to stop you keeping your married name. Many professional people including teachers and solicitors keep their maiden name.
In fact, if you decided to change to your married name, you could always go back to your maiden name in the future.
The government have website: https://www.gov.uk/changing-passport-information/overview
which explains a little more detail.
Can I clarify anything for you?
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