Cool Italian banker’s thick-skinned customer (5)
I believe the answer is:
hippo
'thick-skinned customer' is the definition.
(thick-skinned animal)
'cool italian banker's' is the wordplay.
'cool' becomes 'hip'.
'italian banker' becomes 'po' (Italian river, something that 'banks').
'hip'+'po'='HIPPO'
(Other definitions for hippo that I've seen before include "Aquatic creature" , "Water-loving mammal" , "Great beast" , "Semiaquatic animal" , "Augustine's place" .)