Game up in Italian river, endlessly deep (4)
I believe the answer is:
polo
'game up' is the definition.
(I've seen this before)
'italian river endlessly deep' is the wordplay.
'italian river' becomes 'po' (river in Italy).
'endlessly' means to remove the last letter.
'deep' becomes 'low' (similar in meaning).
'low' with its last letter taken away is 'lo'.
'po'+'lo'='POLO'
'in' acts as a link.
(Other definitions for polo that I've seen before include "Horse-game played with loop" , "Horsey ball game" , "Venetian 19 to Asia, servant to Kublai Khan" , "car?" , "A mint on horseback?" .)