Do any EU countries allow unmarried mothers of British children to live there, without being married to the father?

If a non-EU citizen mother gives birth to a child in Ireland (on a tourist visa) to a man who is British, the child would become both Irish and British. On this basis, could the mother become a resident (and then citizen) of either Ireland, the UK or any other EU member-state, without being married to the father of the child?

Have the parents lived together for two years in a durable relationship?

It would not be provable, so no.

I think there's maybe Zambrano but doesn't lead to residency - that's for EU route, for UK with Brexit etc, thing are only gonna get tighter.

Wait for others.