It's more the other way around. The Romans called the isles Britannia Major and Britannia Minor (now Great Britain and Ireland). Some time after Rome fell, Britons started to settle in another region they called Armorica, so it changed name to Brittany. "Small Britain" could be either Ireland or Brittany at this point.
https://en.wikipedia.org/wiki/Brittany#Etymology
https://en.wikipedia.org/wiki/Britain_(place_name)