See what Argentina is doing to make Britain's life miserable vis-a-vis drilling in the Falklands (or Malvinas). My guess is that they have the right of this, if they don't want ships sailing from Argentina to the Falklands with cargoes of oil drilling supplies, they can probably prevent it.
The Brits can drill but will have to provision the drilling site from somewhere other than Argentina, which makes it more difficult and more expensive. Brazil or South Africa are both possible supply bases.
