County Durham's first store in trouble (6)
I believe the answer is:
dorset
'county' is the definition.
(I know that Dorset is a historic county of England)
'durham's first store in trouble' is the wordplay.
'durham's first' becomes 'd' (1st letter of 'durham').
'in trouble' indicates an anagram.
'store' with letters rearranged gives 'orset'.
'd'+'orset'='DORSET'
(Other definitions for dorset that I've seen before include "South coast county" , "South-west area" , "English county -- sorted (anag)" , "area of Portland Bill" , "A SW county" .)