Population genomics offers innovative approaches to test hypotheses related to the source and timing of introduction of invasive species. These approaches are particularly appropriate to study colonization of island ecosystems. The brown rat is a cold-hardy global invasive that has reached most of the world's island ecosystems, including even highly isolated archipelagoes such as the Faroe Islands in the North Atlantic Ocean. Historic records tell of rats rafting to the southern island of Suðuroy in 1768 following a shipwreck off the coast of Scotland, then expanding across the archipelago. We investigated the demographic history of brown rats in the Faroes using 50,174 SNPs. We inferred three independent introductions of rats, including to Suðuroy, the islands of Borðoy and Viðoy, and onto Streymoy from which they expanded to Eysturoy and Vágar. All Faroese populations showed signs of strong bottlenecks and declining effective population size. We inferred that these founder events removed low frequency alleles, the exact data needed to estimate recent demographic histories. Therefore, we were unable to accurately estimate the timing of each invasion. The difficulties with demographic inference may be applicable to other invasive species, particularly those with extreme and recent bottlenecks. We identified three invasions of brown rats to the Faroe Islands that resulted in highly differentiated populations that will be useful for future studies of life history variation and genomic adaptation.