The Falkland Islands are British and have been for a long time. A country, a city, a person - no entity relinquishes its right or claim to something simply because another entity is closer to it. The British have never renounced their claim to the Falklands Islands and all claims to them having done so are incorrect.
Perhaps I misheard, but there was a 3-month war back in 1982 over the Falklands between Britain and Argentina, and it ended with the Argentineans surrendering in defeat. While Argentina may have made some patriotic noises regarding these islands and their invalid claim to it recently, the fact of the matter is that the war settled the matter and Britain has never renounced its claim.