County Durham's first store in trouble (6)
I believe the answer is:
dorset
'county' is the definition.
(I know that Dorset is a county)
'durham's first store in trouble' is the wordplay.
'durham's first' becomes 'd' (1st letter of 'durham').
'in trouble' is an anagram indicator.
'store' anagrammed 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" .)