English Channel beachfront with bishop having ice cream serving (3,3)

I believe the answer is:

bbc one

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'bishop having ice cream serving' is the definition.
I know nothing about this answer so I can't tell whether this works.

'english channel beachfront' is the wordplay.
'english channel' becomes 'bbc' (I've seen this in another clue).
'beachfront' becomes 'one' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'bbc'+'one'='BBC ONE'

'with' acts as a link.

Can you help me to learn more?

I've seen this clue in the Sydney Morning Herald.

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