Antarctica—0.00008 people per sq. km! McMurdo is by far the largest community on Antarctica with 1200 people in the summer and 250 in the winter. It’s located at the southernmost ice pack-free point on the continent. King George Island is home to a dozen smaller stations whose populations add up. In many cases the stations are so close to one another that they are basically one community (e.g. Fildes Peninsula).
I made this map by getting station populations from Wikipedia (or in some cases other sources), then just joining them to a grid (unequal area, based on the polar orthographic projection) and plotting the centroids of those joined grid cells.
I learned about Antarctic oases, areas that are inexplicably free of ice and snow. Looks a lot like Death Valley, only 120º F colder: https://en.wikipedia.org/wiki/Antarctic_oasis