Final answer:
The British overseas territories around 300 miles off the coast of Patagonia are the Falkland Islands, consisting of East and West Falkland along with smaller islands, known for their subpolar oceanic climate and being the subject of the 1982 Falklands War.
Step-by-step explanation:
The British overseas territories that lie approximately 300 miles off the coast of Patagonia are the Falkland Islands. The archipelago consists of two main islands, East Falkland and West Falkland, along with smaller islands. This region experiences tundra-like conditions, with treeless plains and a cool climate. The Falkland Islands are a self-governing territory under British control, despite Argentina's claim to the islands. This dispute famously escalated into the Falklands War in 1982 when Argentina's military junta attempted to seize the islands, but the British navy intervened and retained control.