One kills animal -- a small offence? (8)
I believe the answer is:
assassin
'one kills' is the definition.
'animal a small offence?' is the wordplay.
'animal' becomes 'ass' (term for a donkey).
'small' becomes 's' (abbreviation - e.g. clothes size).
'offence?' becomes 'sin' (a sin is an offence or transgression).
'ass'+'a'+'s'+'sin'='ASSASSIN'
(Other definitions for assassin that I've seen before include "Killer of prominent person" , "Professional, often political, killer" , "Hitman" , "hit man" , "One murders for political or monetary reasons" .)