A Banbury-man—i.e. a Puritan (Ben Jonson); a bigot.
From the reign of Elizabeth to that of Charles II. Banbury was noted
for its number of Puritans and its religious “zeal.”
As thin as Banbury cheese.
In Jack Drum's Entertainment we read, “You are like a
Banbury cheese, nothing but paring;” and Bardolph compares Slender to
Banbury cheese (Merry Wives, i, 1). The Banbury cheese is a rich
milk cheese about an inch in thickness.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894