I think there are rules against that sort of thing............... I think Argentinians can only get Visitors passports of limited duration - certainly up to 30 days but not sure beyond that
The FI has a Constitution - one article says:-
"for the imposition of restrictions on the movement or residence in the Falkland Islands or on the right to leave the Falkland Islands of persons generally or any class of persons that are reasonably required in the interests of defence, internal security, public safety, public order, public morality or public health, except so far as that provision or, as the case may be, the thing done under its authority is shown not to be reasonably justifiable in a democratic society;"