Finding this lighthouse involved first wondering if there are lighthouses in Antarctica (there are coastal lights, but not really lighthouses), and then looking for a lighthouse on South Georgia Island (unsuccessful), and then trying the Falkland Islands. And there are a few.
This one is fairly close to the largest town on the Falkland Islands, Stanley. This is where it's located. I also found the nice graphic image below for those in the audience who might not know where the Falklands are.
Now that we've seen the artistic version, let's learn more about it via Lighthouse Directory information.
"1907 (station established 1855). Inactive since 1982. 21 m (69 ft) round cast iron tower with lantern and gallery, painted black with a broad white horizontal band below the lantern. The 3rd order Fresnel lens installed in 1907 was smashed by vandals in 1982. The active light (1987; focal plane 30 m (98 ft); three white flashes every 20 s) has been moved atop a short post nearby ... Prefabricated in London, the original lighthouse was an 18 m (60 ft) cast iron tower. It was meant to warn ships away from the Billy Rock, a very dangerous reef about 800 m (1/2 mi) offshore. The original foundation deteriorated seriously and in 1906-07 the current lighthouse was built on a new foundation about 180 m (200 yd) west of the original location. The lighthouse was deactivated after being damaged in the 1982 war between Britain and Argentina. ... In recent years volunteers have worked to restore the lighthouse as a tourist attraction and historical monument.The 150th anniversary of the lighthouse was celebrated on December 1, 2005."
I will note that in searching for pictures, many of them showed the lighthouse with rust stains on the white band. At some point that was either cleaned or painted over (maybe both), because they are also pictures of the lighthouse looking brighter, without the stains. I've got both "versions" below.