One not well wrapped in faded mattress (9)

I believe the answer is:

paillasse

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'mattress' is the definition.
(paillasse is a kind of mattress)

I don't understand the rest of the clue.

(Other definitions for paillasse that I've seen before include "straw bed" , "Maybe sleep on it" , "Bedding" .)

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

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