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Would she be required to get a seperate visa to enter France?
Yes. The UK and France are separate territories for the purpose of immigration control, and they have different visa exemption lists. She only needs a visa, though, if her passport is from a so-called "Annex I" country. If she has an "Annex II" passport, she does not need a visa for the Schengen area. If you are a British citizen or a citizen of another EU or EEA country, and the trip takes place before the UK leaves the EU free movement scheme, the visa should be free of charge.
And will there be any issues re-entering the UK?
Generally she should have no problem returning to the UK. Her BRP is sufficient to show that she is authorized to travel to and enter the UK.
I say "generally" because the standard disclaimer applies: the legal possibility of refused entry continues to exist, but only in exceptional cases such as the discovery that the holder if the BRP has committed some serious crime.