Only a fathead can keep baby quiet! (5)

I believe the answer is:

dummy

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'baby quiet' is the definition.
The answer and definition can be both people as well as being singular nouns.
Perhaps there's a link between them I don't understand?

I don't understand the remainder of the clue.

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(Other definitions for dummy that I've seen before include "Make as if to pass" , "that all the players can see" , "who lays his cards on the table?" , "Bridge hand on table" , "Mock-up; non-working version" .)

I've seen this clue in the Evening Standard.

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