It's a bit more complicated than a simple this-or-that distinction, i'm afraid. Old-Norman-French/Old-Northern-French - the language most commonly associated with the Norman peoples of Northern France, the Anglo-Norman royalty post-invasion of England, and the Norman nobility of the early Kingdom of Two Sicilies was a Romance (latin root) language that adopted a wide range of Norse vocabulary, phonology, and even basic syntax. One could go so far to say by the very nature of language in the medieval era what would count as "french" would vary from region to region. Needless to say, all of this greatly complicates the question at hand.
So to answer this as simply as possible; Norman lands were far from homogeneously French-Speaking and in fact would have contained regional french dialects, "royal french" (that spoken by aristocracy), Old-Norman-French (which would have dominated courts and major towns), and a spattering of old-Norse dialects (though almost certainly the rarest.) Remember that the Norse never replaced anywhere near the majority of the local french populous - most people would have been born into purely French, or French-Norse ancestry. The French language only began to homogenize with the expansion of printing and common media - which may be argued to have been "completed" as late as the 18th or even 19th century.
To summarize, most Norman communities of 1066 would have either spoken the old-Norman-French hybrid or a basic french regional dialect. Most Norse would be a second language.
As a rule of thumb for questions of dialect, it is better to lean away from the "homogeneous dialect" answer, as language is too dynamic to ever be considered anything but dialectical - at least until the information age.
Check out Emily Albu's The Normans and their Histories And Richard Ingham's The Anglo-Norman Language and Its Contexts, introductory chapter and chapter 2
There's no strong evidence either way, but there are hints that one or more Scandinavian languages may have persisted to some extent for longer than we might assume. (I say 'Scandinavian' here because it's been suggested that the settlers who became the Normans were largely Danish, led by a handful of Norsemen).
Lauren Wood Breese argues in her article The Persistence of Scandinavian Connections in Normandy in the Tenth and Early Eleventh Centuries that there were Scandinavian loan words present in Norman French and that regular visits to Normandy by Scandinavian traders and pirates would have made retention of the language advantageous. Dudo of Saint-Quentin mentions Duke Richard I of Normandy being sent to Bayeux to learn the language of his forebears but Dudo isn't the most reliable source, being basically a pro-Norman propaganda machine. While we're on the topic of Richards, Breese also mentions 'a visiting skald entertained the court of Richard II [of Normandy] with verses not likely sung in Old French'.
The key thing to note here is that even if Breese is right, we're dealing with the decades preceding William the Bastard's invasion of England. While fifty years or so is nothing in historical terms, it's plenty of time to let a language slide into disuse. Perhaps more importantly, though, the method of conquest that the Normans used tended to involve blending in with the conquered - adoption of the languages and administrative structures in the conquered territories. Personally I think there's a good chance that whatever Scandinavian linguistic remnants persisted in Normandy, they probably didn't survive in England after the conquest (though having said that, there was a substantial Danish population in England, so maybe spoken Danish would be an advantage).
I think it's worth pointing out that there's a huge difference between Norman England and Normandy itself. It's entirely possible that there was no Scandinavian spoken in England but still some spoken in parts of Normandy. Breese feels that the Cotentin peninsula was a good candidate for the place most likely to hang onto its Scandinavian roots, since there's some indication that Scandinavian burial practice persisted there for a surprisingly long time.
Oddly enough, Prof. Jeremy Johns makes a passing mention in his short article The Language of Islamic Art that 'a handful of Normans [in Sicily] spoke Norse'. The Hauteville family who settled Sicily originated in the Cotentin region of Normandy, so that would fit with Breese's belief that the Cotentin was unusually Scandinavian. Johns doesn't offer any support for his claim, though, and when I emailed a couple of years ago to enquire, his response amounted to basically 'well, I think a few probably did. It's possible.' Oddly sloppy, considering how meticulous Prof. Johns' work usually is. I have a suspicion that there's evidence but he doesn't want to share it until he's published an article or book chapter about it.
The TL;DR version is: there's very little evidence that Norse (or any other Scandinavian language) persisted in Normandy but there are logical reasons to think the Normans might have made an effort to preserve it. By the time the conquest of England rolled around, though, there probably wasn't much to be gained by hanging onto any Norse roots and there's certainly no indication that the language was spoken.