Mum's keeping mum in solicitors (6)

I believe the answer is:

madams

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'solicitors' is the definition.
The definition and answer can be both people as well as being plural nouns.
Perhaps you can see a link between them that I don't see?

'mum's keeping mum' is the wordplay.
'mum' becomes 'ma' (both can mean mother).
'keeping mum' becomes 'dams' (I can't justify this - if you can you should believe this answer much more).
'ma'+'dams'='MADAMS'

'in' acts as a link.

Can you help me to learn more?

(Another definition for madams that I've seen is " Precocious girls".)

I've seen this clue in The Guardian.

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