Lady Diana's first university game (7)
I believe the answer is:
duchess
'lady' is the definition.
(I know that duchess is a type of lady)
'diana's first university game' is the wordplay.
'diana's first' becomes 'd' (1st letter of 'diana').
'university' becomes 'u' (abbreviation for university).
'game' becomes 'chess' (chess is a kind of game).
'd'+'u'+'chess'='DUCHESS'
(Other definitions for duchess that I've seen before include "Titled female" , "Titled lady" , "Great lady" , "Duke's wife" , "Noble lady" .)