Dog to slip inside bank (7)
I believe the answer is:
terrier
'dog' is the definition.
(terrier is a kind of dog)
'slip inside bank' is the wordplay.
'slip' becomes 'err' (to err is to make an error).
'inside' means one lot of letters goes inside another.
'bank' becomes 'tier' (both can mean a row of similar things).
'err' going inside 'tier' is 'TERRIER'.
'to' is the link.
(Other definitions for terrier that I've seen before include "Breed of dog - volunteer soldier - estate register" , "See 11" , "Dog originally used to ferret out burrowing animals" , "(Yorkshire?) dog" , "Dog - reservist" .)