Buoy up (9)
(Other definitions for lift that I've seen before include "Raise - shot in the arm" , "It's an elevator in USA" , "Pick up" , "Raise, elevate" , "Raise - hoist" .)
'buoy up' is the definition.
(thesaurus)
This is the entire clue.
(Other definitions for encourage that I've seen before include "Give support to, stimulate hope" , "Hearten, inspire" , "Hearten, foster" , "Give hope to (someone)" , "Cheer" .)