Found this citation:
ASHBURNHAM or as Camden calls it Esuburnam is a parish in Foxearl hundred Hastings rape 7 miles from Battle and 61 from London containing 94 houses and 786 inhabitants From this place a very ancient family derives its name Bertram son of Anchitel and grandson of Piers de Ashburnham was constable of Dover Castle when William duke of Normandy invaded this country and was beheaded for holding out that fortress against him John de Ashburnham was knight of the shire in the reigns of Richard II and Henry IV His son and grandson were in their turns Sheriffs of Surrey and Sussex Their descendants were singularly loyal to king Charles I John Ashburnham in particular was one of the first who took up the cause of the dethroned sovereign and having afterwards supplied king Charles II with considerable sums of money was taken by Cromwell and committed to the tower After the Restoration he was elected groom of the bed chamber to the king and returned knight of the shire for this county in 1661
https://books.google.com/books?id=qlBgAAAAcAAJ&pg=PA41&lpg=...
Considering how *very* late the source is, a fair amount of salt is in order. "Piers", "Anchitel", and "Bertram" are all Norman names - not Saxon.
Domesday Book records the following for Ashburnham:
Lord in 1066: Siward.
Overlord in 1066: King Edward.
Lord in 1086: Robert of Criel.
Tenant-in-chief in 1086: Robert Count of Eu.