Yes, it was the Mayflower John (who was hung)'s DAD who was listed as Billington "Bullington".I haven't been able to find out how someone got that far, though (haven't found any sources further back than the Mayflower).
If you go to http://www.familysearch.org/Eng/Search/frameset_search.asphttp://www.familysearch.org/Eng/Search/frameset_search.asp then enter Francis (1st name) Billington (last name), mark the event as Birth/Christening, in the next space mark Exact Year, and for the year put 1606, and the country as England, it will bring up only one Francis under "Ancestral File."If you click on his name, the next screen has "Pedigree" and "Family" along the right side.I clicked on the Pedigree, and could see his father John and grandfather John (possibly Bullington).All the info I've read from the Mayflower (passenger list, Gov Bradford's accounts, etc.) refer to the Mayflower ones as Billington or Billinton.
Francis did have several children, but I descend through one of his daughters, so don't know about any of the Billington (or possibly Bullington) descendants after him.