Hospital department follows each person full-time (9)
I believe the answer is:
permanent
'full-time' is the definition.
(I've seen this before)
'hospital department follows each person' is the wordplay.
'hospital department' becomes 'ent'.
'follows' means one lot of letters go next to another.
'each' becomes 'per'.
'person' becomes 'man' (man is a kind of person).
'per'+'man'='perman'
'ent' after 'perman' is 'PERMANENT'.
(Other definitions for permanent that I've seen before include "Lasting, durable" , "Lasting, unchangeable" , "This way for the track" , "Unchanging" , "Lasting; not temporary" .)