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