The dogs ran wild, being wilful (10)
I believe the answer is:
headstrong
'wilful' is the definition.
(synonyms)
'the dogs ran wild' is the wordplay.
'wild' is an anagram indicator.
'the'+'dogs'+'ran'='thedogsran'
'thedogsran' is an anagram of 'HEADSTRONG'.
'being' acts as a link.
(Other definitions for headstrong that I've seen before include "Showing wilfulness" , "Obstinate" , "strong-willed" , "pig-headed" , "Wilfully determined" .)