There is indeed oil there:
http://www.rigzone.com/news/article.asp?a_id=114761
"Any success by the firm would likely be significant since no oil has yet been found in the South Falklands Basin, unlike in the zone north of the Falkland Islands where fellow UK firm Rockhopper Exploration made an oil discovery on its Sea Lion target in 2010".
Above is a quick, current, article on the state of play in the Falklands. A new well is being started on a new potential prospect but oil has already been found in a zone north of the Falkland Isles......
http://www.rockhopperexploration.co.uk/
"In February 2010, the Ocean Guardian drilling rig arrived in Falklands waters to carry out a multi-well drilling campaign. Rockhopper drilled an exploration well on its Sea Lion prospect during April and May 2010, the result of which is the first oil discovery and Contingent Oil Resource in the North Falkland Basin".
"With the well on the Sea Lion prospect successfully flow tested during September 2010, the company is fully funded to undertake an extensive exploration and appraisal programme across all of its acreage during 2011".
