Dampen down fire initially in wooden gate (6)

I believe the answer is:

stifle

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'dampen down' is the definition.
'stifle' can be an answer for 'dampen' (synonyms). I'm unsure of the 'down' bit.

'fire initially in wooden gate' is the wordplay.
'initially' suggests taking the first letters.
'in' is an insertion indicator.
'wooden gate' becomes 'stile' (I've seen this in another clue. I am not sure about the 'wooden' bit.).
The initial letter of 'fire' is 'f'.
'f' going into 'stile' is 'STIFLE'.

Can you help me to learn more?

(Other definitions for stifle that I've seen before include "Suppress of choke" , "Smother or suppress a feeling or action" , "Gag" , "Quash" , "Smother, check" .)

I've seen this clue in The Times.

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