Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
ANNE
As described in historical dictionaries
Patronymica Britannica (1860)
ANNE.* The name is local. The pedigree begins with Sir William de Anne, Constable of the castle of TickJiill, co. York, temp. Edward II. Shirley's Noble and Gentle Men.