I've never thought that the Irish were British: cf the United Kingdom of Great Britain and [later: Northern] Ireland.
Now Marcellous is right and then again he is not!
Until I read Norman Davies I had no idea just how complicated things were. I should do a post drawing from Davies on the changing title of that constitutional entity currently known as the UK.
Note to Readers: for a full list of posts in this series see Train Reading - Norman Davies The Isles: a history 1.