London — Adding to her EU divorce dramas, Prime Minister Theresa May is heading to the G-20 summit in Argentina, a country using Brexit to advance its claim on Britain’s Falkland Islands.

The visit, the first by a British premier to the Argentinian capital since the 1982 Falklands War between the two countries, is being seen in Buenos Aires as a chance to improve ties.

However, the 3,400 Falkland Islanders insist their British sovereignty remains non-negotiable and say a proper thaw in their relations with Argentina remains distant.

They have reason to be worried about Brexit: if Britain leaves the EU without a deal, the Falklands' economy — heavily reliant on tariff-free squid exports to EU member Spain — could face a hammer blow.

Argentinian foreign minister Jorge Faurie made waves in October, saying Buenos Aires would exploit Brexit to enhance its diplomatic push for the Falklands.

Britain's sovereignty claim to the islands, known in the Spanish-speaking world as the Malvinas, dates back to 1765 and it has held permanent administration since 1833. Buenos Aires claims the barren, windswept islands, 400km  from the Argentinian coast, are occupied  territory.

Argentina’s then-ruling military junta invaded on April 2 1982 but surrendered on June 14 to a British task force after a brief but intense and bloody war.

Brexit is also an opportune moment for Anglo-Argentine ties: Britain is seeking new trade partners outside the EU while Argentina needs to find ways out of an economic crisis.

Faurie  said in New York in November that Buenos Aires was discussing with London the possibility of "creating more connectivity to the islands".

"There is a lot of room still to grow in terms of confidence and trust between the two countries," he said.

"Our expectation is that the rebuilding of bilateral trust ... will constitute a sort of substantive material to discuss all other issues about the sovereignty of the Malvinas."

Once Britain is outside the EU in March 2019, the 27 other states will no longer be obliged to support London’s sovereignty position.

Brexit is also an opportune moment for Anglo-Argentine ties: Britain is seeking new trade partners outside the EU while Argentina needs to find ways out of an economic crisis.

"Post-Brexit, Argentines are imagining that if the Falkland Islands has to get its economic act in order, that may open up channels," Richard Lapper, a South America specialist at Chatham House, the Royal Institute of International Affairs think-tank in London, said.

"The big prize for Britain is the prospect of very cheap agricultural imports.The Argentines may say they only want to do a deal on food if we have a side agreement on the Falklands," Lapper said.

Jimena Blanco, Americas chief and Argentina expert at risk analysts and strategic forecasters Verisk Maplecroft, said President Mauricio Macri had moved away from the Falklands "sabre-rattling" of his nationalist predecessor Cristina Kirchner.

Macri's Falklands policymakers "have a more moderate approach", she said.

"There is an understanding that the war, and confrontation under Kirchner, made a lot of damage that cannot be undone by one administration.

"It needs to become a sustained state policy whereby future generations of Falkland Islanders no longer see Argentina as a threat but more as a neighbour that they might want to have a special relationship with."

As an EU member state's overseas territory, the Falkland Islands enjoy tariff- and quota-free access to the European single market. About 94% of fisheries exports go to the EU, contributing 40% of the islands' GDP and a third of government revenue. The second-biggest sector is meat and wool exports.

It is estimated that a no-deal Brexit would shrink the Falklands government's revenue by up to 16%, damaging its ability to deliver public services and invest in roads and hospitals.

Tariffs would "wipe out the EU as a market for meat exports", while there "really isn't an alternative market" for its squid than southern Europe, said Richard Hyslop, senior policy adviser to the Falkland Islands government.

"The fishing industry is a huge success story for the Falklands and we don't want to see anything happen that damages that," he said.

Even so, establishing alternative trade links with Argentina seems a long way off. The sovereignty claim is enshrined in the constitution and Kirchner-era legislation aimed at frustrating the Falklands economy remains in place.

"Certainly I don't think anyone's looking at direct trade with Argentina," said Teslyn Barkman, the Falklands MP responsible for Brexit, the EU and natural resources. "Regardless of Brexit, they would be trying to find opportunities to work this sovereignty conversation into any development."