Britisher shouldn't even occur to someone above age 10.
From the forbes.com
Britisher Ken Stott is a veteran stage actor and Reza specialist who created this role in the London premiere.
From the bloomberg.com
More examples
A native or inhabitant of Great Britain
The British (also known as Britons, informally Brits, or archaically Britishers) are citizens of the United Kingdom, of the Isle of Man, one of the Channel Islands, or of one of the British overseas territories, and their descendants. ...
A Briton
(Brit) (AmE) British, usually specifically English, but not derogative.