Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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The Argentine surname in historical dictionaries
Patronymica Britannica (1860)
ARGENTON. ARGENTINE. Argentan, a considerable town in the south of Normandy, formerly written Argcntomagus. David de Argentomago Avas a tenant in chief under the Conqueror, in cos. Bedford and Camb. His descendants were ennobled as barons Argentine.