The phrase building wing refers to a specific section of a building. A building that is shaped like an L would have 2 different wings.
A building's extension is a wing. As in, "The hospital is adding a wing". Or, "They named it 'The Gonzo Wing' because Gonzo funded it".
If you mean the side arms of a church, it is called "the transept". Each of the two may contain a chapel.
West Wing
It is just called the wing.
If you are referring to the distance from wing tip to wing tip, it is usually called "wingspan".
Annex
the wing flap
ell
The White House
it's a part of a building designated for remixing tunes, often referred to as "the remix wing". example: Where is Dave? Oh, he's over in the remix wing.
the south wing