There's a previous answer for your question : what became Normandy was one of the two Marchs of Neustria.

The transition doesn't seem to have been immediate or overlapping : while the name undoubtedly comes from the medieval latin Normanni, it appears in the mid-to-late Xth century to name the territories ruled by Rollo and his descendents as counts/marquess/duke "of the Normans" and including the coastal regions settled by Anglo-Scandinavians or Hiberno-Scandinavians rather than Danes, with inner regions as well that were largely not so into a same broad political identity.