The garrison is at Mount Pleasant which almost certainly isn't. You can read all about it in today's Sunday Telegraph.
I can just about understand why anyone would want to live in the Falkland Islands; I can't understand why "we" want to keep the Islands. The inhabitants don't contribute anything to the British Exchequer but "we" spend vast amounts on defending them from the Argie threat. The Falklands are a loss-making subsidiary and "we" have more than enough of those already (Northern Ireland, Wales).
When the Germans occupied the Channel Islands in World War Two, "we" didn't rush to kick them out: the Channel Islands are not part of the United Kingdom and "we" were rather more concerned to defend the UK from its enemies. That bears remembering: all that money spent on the Falklands could be spent undermining the appeal of terrorism to people living in the UK.
My suggestion is this: since the Argentinians are so keen to own the Falklands, why not let them buy the Freehold for a deficit-busting amount - and on condition that they grant a 99 (or 999 year) lease to the existing tenants.
After all, Imperial Russia did not die of shame when it sold Alaska to the Americans, even though Alaska is on Russia's doorstep rather than America's.
My suggestion would also work well for Gibraltar - though in the past I have supported the idea of offering Hastings to Spain in an exchange deal. It would really transform Hastings into a great place for a day out.