Elite military band cut short (3)
I believe the answer is:
sas
'elite military' is the definition.
(Special Air Service in UK army)
'band cut short' is the wordplay.
'band' becomes 'sash' (a band of material).
'cut short' means to remove the last letter.
'sash' with its last letter taken away is 'SAS'.
(Other definitions for sas that I've seen before include "Elite fighting force" , "Specialist regiment of the British army, in short" , "Commando-type unit of the British army" .)