Rover's given up on barking (5)
I believe the answer is:
nomad
'rover's' is the definition.
(nomad is a kind of rover)
'up on barking' is the wordplay.
'up' is a reversal indicator (in down clue: letters go upwards).
'barking' becomes 'mad' (barking mad).
'on' reversed gives 'no'.
'no'+'mad'='NOMAD'
'given' acts as a link.
(Other definitions for nomad that I've seen before include "someone who can't settle" , "Roving person" , "Person always on the move" , "Unsettled type" , "One keeps moving" .)