This area's for walking dog, said policeman brooking no nonsense (4,8)
I believe the answer is:
peak district
'this area's for walking' is the definition.
The definition suggests a singular noun which matches the answer.
'dog said policeman brooking no nonsense' is the wordplay.
'dog' becomes 'peke' (short for Pekingese).
'said' shows a homophone (sound like).
'policeman' becomes 'di' (abbreviation for Detective Inspector).
'brooking no nonsense' becomes 'strict' (I can't explain this - if you can you should believe this answer much more).
'peke' sounds like 'peak'.
'peak'+'di'+'strict'='PEAK DISTRICT'
Can you help me to learn more?
(Other definitions for peak district that I've seen before include "South Pennines Park" , "Region of north Derbyshire" .)