A part of church that liberal’s left, possibly (9)

I believe the answer is:

perchance

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'possibly' is the definition.
(synonyms)

'a part' is the wordplay.
'a' becomes 'per' (eg 'one a day' means 'one per day').
'part' becomes 'chance' (I have seen 'part of Monopoly' mean 'chance' so perhaps 'part' could also mean 'chance').
'per'+'chance'='PERCHANCE'

'of church that liberal's left' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for perchance that I've seen before include "possibly" , "As it may happen" , "Maybe (to dream? - Hamlet)" , "'To sleep: -------- to dream:' Hamlet" .)

I've seen this clue in The Sunday Times.

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