What is the distance between the most southerly part of South America and the most northerly part of Antarctica?

About 1,000 kilometers.

https://en.wikipedia.org/wiki/Antarctic_Peninsula

1000km / 600 miles

Antarctica is the fifth largest of the seven continents. It is situated over the South Pole almost entirely south of latitude 66°30' south (the Antarctic Circle). It is a very rough circular shape with the long arm of the Antarctic Peninsula stretching towards South America. There are two large indentations, the Ross and Weddell seas and their ice shelves.

The nearest other land masses are South America 600 mls / 1000 km away across the roughest stretch of water in the world - the Drake passage, Australia is 1550 mls / 2500 km away, and South Africa 2500 mls / 4000 km away.

The total surface area is about 14.2 million sq km (about 5.5 million sq mls) in summer, approximately twice the size of Australia, half as big again as the USA and fifty times the size of the UK. In the winter Antarctica doubles in size due to the sea ice that forms around the coasts. The true boundary of Antarctica is not the coastline of the continent itself or the outlying islands, but the Antarctic Convergence.

http://www.coolantarctica.com/Antarctica%20fact%20file/antarctica%20environment/whats%20it%20like%20in%20Antarctica.htm