Maritime surveillance of the Arctic region is of growing importance as shipping, fishing and tourism are increasing due to the sea ice retreat caused by global warming. Ships that do not identify themselves with a transponder system, so-called dark ships, pose a security risk. They can be detected by SAR satellites, which can monitor the vast Arctic region through clouds, day and night, with the caveat that the abundant icebergs in the Arctic cause false alarms. We collect and analyze 200 Sentinel-1 horizontally polarized SAR scenes from areas with high maritime traffic and from the Arctic region with a high density of icebergs. Ships and icebergs are detected using a continuous wavelet transform, which is optimized by correlating ships to known AIS positions. Globally, we are able to assign 72% of the AIS signals to a SAR ship and 32% of the SAR ships to an AIS signal. The ships are used to construct an annotated dataset of more than 9000 ships and ten times as many icebergs. The dataset is used for training several convolutional neural networks, and we propose a new network which achieves state of the art performance compared to previous ship–iceberg discrimination networks, reaching 93% validation accuracy. Furthermore, we collect a smaller test dataset consisting of 424 ships from 100 Arctic scenes which are correlated to AIS positions. This dataset constitutes an operational Arctic test scenario. We find these ships harder to classify with a lower test accuracy of 83%, because some of the ships sail near icebergs and ice floes, which confuses the classification algorithms.