Small argument was formal (5)
I believe the answer is:
'formal' is the definition.
(similar in meaning)
'small argument' is the wordplay.
'small' becomes 's' (abbreviation - e.g. clothes size).
'argument' becomes 'tiff' (I've seen this in other clues).
'was' acts as a link.
(Other definitions for stiff that I've seen before include "18 [HARD]" , "cheat" , "One dead" , "Rigid, unbending" , "Starched, rigid" .)