Yes, all foreigners can live in Svalbard without a visa.
However, you’ll probably need to have housing and also a double-entry visa to re-enter Norway upon leaving Svalbard.
From the Governor of Svalbard’s site:
Visa to Schengen
In mainland Norway, the Immigration Act regulates foreigners’ access to and residence in the country. Although Svalbard is part of Norway, the Immigration Act does not apply to the archipelago.
Foreigners do not need a visa or work and residence permits from the Norwegian authorities to travel to Svalbard. However, foreign citizens with a visa requirement for the Schengen Area must have a Schengen visa when travelling to and from Svalbard via mainland Norway. It’s important to ensure that you get a double-entry visa so you can return to the Schengen Area (mainland Norway) after your stay in Svalbard.
Requirements for staying
Although you do not need a visa or your own work and residence permit, everyone must meet certain requirements in order to stay in Svalbard. These requirements are governed by a separate policy called “Regulations relating to rejection and expulsion of persons from Svalbard”. Among the requirements is that you must have the means to be able to reside on Svalbard. These requirements apply to both foreigners and Norwegian citizens, and The Governor of Svalbard may reject persons who do not meet the requirements. Therefore, those planning to come to the archipelago are advised to obtain work and housing before they arrive.
(emphasis, apart from the titles, is mine)