I dont think you would want to call an Irishman "British".
This is my understanding of all that stuff:
Great Britain - England, Scotland, Wales
United Kingdon(UK) England, Scotland, Wales & Northern Ireland (not the Isle of Man or the Channel Islands)
British Islands - United Kingdom, the Channel Islands and the Isle of Man.
British Isles - Britain, Ireland and many smaller surrounding islands
So in a sense, one could say an Irishman is rom the British Isles - but call one British and I imagine you will get a good reaction.
Its funny, that those from Europe use Yank as a general slang term. Whereas in the States, those Americans from the South, use Yank as a pretty strong derogatory term for those Americans from the Northern States. I remember working in the South once and getting a nasty phone call once from a very friendly gentleman who threatened to shoot me down with his rifle. "You aint from round here, are ya boy?" "Nope!", I replied. "I knew it, ya goddamn Yankee!" lol - very entertaining, asides from the fact he wanted to shoot me!