The origin of the name:
"Bayman is a corruption of Baynham to which there are many parallels...this surname is still familiar to co. Wilts. bordering
on the district where Baynham arose. There cannot be a doubt as to the origin. "
(Dictionary of English Surnames, Bardsley G.W.)
"Ab Eignon (Welsh) i.e.'the son of Eignon' or 'Ennion'. A Gloucestershire surname.Robert ap Eignon had for his son Robert Baynham of Chorewell, the Forrest of Dean. Henceforth this family were so known (vide Visitation of Gloucestershire, 1623, p 14; Harleian Society). The name looks wonderfully English and local, but, as shown, is not so.
Beynon, Binyon and Benyon, q.v., are other forms; but not Bunyan, I think.
(Dictionary of English & Welsh Surnames, Bardsley, G.W. 1980)