Enigma about green light bathing dance band's oily faces (6,4)
I believe the answer is:
closed book
'enigma' is the definition.
(I've seen this before)
'about green light bathing dance band's oily faces' is the wordplay.
I cannot quite see how this works, but
'about' could be 'c' (circa) and 'c' is located in the answer.
'band's' could be 'os' (I've seen this in other clues) and 'os' is found within the answer.
'light' could be 'led' (light-emitting diode) and 'led' is located in the remaining letters.
The remaining letters 'book' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for closed book that I've seen before include "Mystery person" , "Subject about which one knows nothing" , "Took no more bets" , "Enigma" , "What loss-making turf accountant did" .)