Absolutely not. Much more of this 'Little England' mentality and we will become a joke of a country with no status, no trade and no influence anywhere other than in Scunthorpe!
I think if this question were to be asked by someone who lives and works in another country, we might think it's just trolling, wouldn't we?
As an ex-pat who only holds a British passport I'm clearly biased...but no way. A few months ago if someone asked me if I'd apply for a French passport, I'd have said absolutely no...no point. Brexit has changed that and now I'm considering applying for a French one, to keep an EU passport and protect my right to reside in France. As RTID pointed out...suggesting you have only a British passport, or you have to give it up...you might find a lot of people chose to give it up in favour of a passport from a country that is still a part of the EU.
Don't know if you have been in France a long time Dave but if you been there from before free movement came into effect early Nineties I think, you may have permanent / residence rights there from then anyway. Seems I have this in Norway luckily.
I think it's entirely up to the individual whether they choose dual nationality or not. If a person is eligible for such an arrangement why not?...
People who have been resident in another EU country for more than a certain number of years have 'so called' acquired rights guaranteed by the Vienna Convention - which Britain is signatory to and which is external to the EU. Forced repatriation of people who are in possession of de facto legal residence would be a human rights issue. Nobody can imagine a situation similar to the partition of India where we have millions of people forced to uproot themselves, in both directions. The worst that can happen is difficulties in getting qualifications recognized, and also problems getting the full pension paid. Those who were already legally there at the time of the activation of article 50 are there to stay - and cannot be punished retrospectively - and this applies also to EU. citizens living in the UK. As for the issue of dual nationality - Britain recognizes this with most countries in the World, so why not with EU. ones ? I have had dual nationality for many years, but it doesn't actually mean I have 2 nationalities - rather, I am legally German when in Germany, and British when in Britain.
I've been here nearly seven years now and honestly think it'll be fine and I'll be able to stay as I am. If any trouble does come...I may well apply for a French passport though.
Dual nationality can be a practical problem if it involves a state where conscription for national service is still possible - depending obviously on age. A British citizen living in eg. Greece (If under 40) would have to do national service there if they became Greek - dual nationality could not protect them against this because Britain doesn't have it. I have had students in the past who were born in Germany, of Turkish parentage, who had Turkish nationality (though never having lived there) - who then went on holiday with the family to Turkey and their originally planned holiday of 2 weeks became an enforced one of 2 years because of this.