I thought, "Carte de Séjour". They can grant you a ten-year Carte de Séjour which will give the possibility of staying until retirement in the UK.
So I started to research online. Where I found lots of helpful government websites...
It transpires that after living five years in France you can apply for French nationality.
This contrasts interestingly with the UK where, for non-europeans, after five years if you aren't earning over £30,000 a year, home you go. Well-loved teacher? Tough!
Anyway, people applying for French nationality can retain other nationalities.
So, since we have lived in France for 11 years, we can in theory become French, hence European for the remainder of the twilight of the European experiment, and still stay subjects of Her Britannic Majesty and watch her United Kingdom untied.
This sounds like a plan.