woman, they

No.

1) You can't use a singular noun without a preceding determiner - eg

No woman was allowed to...

Alternatively, you could use the plural :
Women were not allowed to ...

2) It makes no sense to use the pronoun "their" with a singular noun which is clearly exclusively feminine. So :

No woman was allowed to use her mind...
or
Women were not allowed to use their minds...

3. their is used as a singular pronoun with neutral gender reference - ie when it could refer to either males or females - eg

A person who has experienced having their house burgled is unlikely to forget it.

but yes - the following noun remains singular. Another example :

No-one will be allowed to change their mind after the decision has been taken.

Excellent answer. Thank you. But a friend of mine, non-native, told me once we could say

-Woman (not women) is the most intelligent creature in the world.

without using the before woman either. Do you think this friend of mine is wrong, or is the sentence she suggested a special use?(=uncountable use of woman)

Yes - here woman is being used uncountably, so the grammar is different. In your example above it was being used as a countable noun. See this thread.

Good answer. Thank you.