One didn't mind a barking dog (6,7)
I believe the answer is:
dandie dinmont
'dog' is the definition.
(Dandie Dinmont is an example)
'one didn't mind a barking' is the wordplay.
'barking' indicates anagramming the letters (UK slang for mad).
'one'+'didnt'+'mind'+'a'='onedidntminda'
'onedidntminda' with letters rearranged gives 'DANDIE DINMONT'.
(Other definitions for dandie dinmont that I've seen before include "Small terrier with drooping ears" , "Dog" .)