Philosopher I support for a very short time (11)

I believe the answer is:

millisecond

'a very short time' is the definition.

'philosopher i support' is the wordplay.
'philosopher' becomes 'mill' (I've seen this before).
'support' becomes 'second' (I've seen this before**).
'mill'+'i'+'second'='MILLISECOND'

'for' is the link.

(Another definition for millisecond that I've seen is " extremely short period of time".)

I've seen this clue in The Times.

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