Bishop, getting fit, is shot in part of church (9)

I believe the answer is:

baptistry

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'in part of church' is the definition.
(I've seen this before)

'bishop getting fit is shot' is the wordplay.
'bishop' becomes 'B' (abbreviation for bishop in chess).
'getting fit' becomes 'apt' (I can't explain this - if you can you should believe this answer much more).
'shot' becomes 'try' (both can mean an attempt).
'b'+'apt'+'is'+'try'='BAPTISTRY'

Can you help me to learn more?

(Another definition for baptistry that I've seen is " In part of church".)

I've seen this clue in The Times and The Sunday Times.

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