It, as well as a good deal of France surrounding what we would recognize as the Duchy of Normandy, including, Paris, Orleans, Anjou, and Blois, was called Nuestria (Western Kingdom) by the Merovingian and Carolingian Franks, as opposed to Austrasia (Eastern Kingom, in what is now considered Germany). Austrasia composed what could be thought of as the traditional homeland of the Franks, while Nuestria was conquered by Clovis, founder of the Merovingian dynasty, shortly after the collapse of the western roman empire, in the fifth century ad. Normandy didn't become Normandy until the tenth century, so the people who were already living there would have spoken Old French and considered themselves French. Even after the arrival of the Northmen, there's very little evidence to suggest that the local population was substantially displaced, far from it. While the original norman settlers/conquerors spoke norse, it didn't take more than a few generations for them to adopt the local language and culture, which again was French, although the northmen did have enough of an impact on the language for Norman French to be it's own dialect for a while.