[dangerface]

People in Northern Ireland like me consider themselves British and we are a third of the population, while obviously not the majority I wouldn't call it uncommon.

[overscore]

I also believe anyone in Northern Ireland who consider themselves to be British are British. However, does that imply usage of the term "British Isles"?

Simply Britain and Ireland (or Ireland and Britain) is far more commonly used if referring to the archipelago.