A close friend beginning to mistrust NATO, for example (7)
I believe the answer is:
acronym
'nato for example' is the definition.
(NATO is an example)
'a close friend beginning to mistrust' is the wordplay.
'close friend' becomes 'crony' (I've seen this before).
'beginning to' says to take the initial letters.
The initial letter of 'mistrust' is 'm'.
'a'+'crony'+'m'='ACRONYM'
(Other definitions for acronym that I've seen before include "eg RADA" , "radar, for one" , "Radar or Nato, maybe" , "Word formed from initial letters of other words." , "NATO's one" .)