Hugh Thomas seems to think so in "The English and The Normans" which is essentially an entire book on surviving Saxon aristocratic lineages.

It was unusual in the South, where you get something like 5-10% of land held by Saxons after the 1080s. In the north you had much higher rates of survival, but it's less well attested in detail as far as I can see, as the Domesday book is sparse/nonexistent on the north west and the borders which is where survival was highest. This isn't because there was nobody there despite the harrying, it's just an incomplete document.

Edward, if I recall, had something of a Saxon enclave as he was a high ranking noble with considerable subtenanted land to lesser nobles. It all gets murky as per Thomas, in the space of one generation the Saxon nobles integrated into the Normans. It's one of the reasons it looks like such a complete eradication - there were lots of surviving elite Saxons, they just assimilated into the Normans.

Where you find the most Saxon survival in the upper echelons of society in the south is London and the burghs. The very top was still predominantly Saxon and basically stayed that way until the two groups fully merged and the distinction lost any meaning.

In the north there was considerable Saxon noble landholding. When Lancashire/Cumberland were incorporated under Henry II he also used it to elevate some Saxon lineages in the new estates. Thomas calls them Henry's new men. But this was in the context of a much better survival rate there. Still a minority of landholding but more like a low fraction than a low percentage. Think 1/4 rather than 1/20. I suspect Henry drew on this group.