Billed character's catchphrase "Where's me washboard?" (4)
I believe the answer is:
smew
'billed character's' is the definition.
(type of duck - ducks have bills)
'catchphrase where's me washboard?' is the wordplay.
I cannot really understand how this works, but
'me' is within the answer.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for smew that I've seen before include "Species of merganser" , "Sea duck" , "Diver" , "Bird" , "Small merganser" .)