Small ship instance segmentation from synthetic aperture radar (SAR) images is a challenging task. Because small ships have smaller scales, indistinct contours, and weak feature response. In addition, background interference and clutter make feature extraction of small ships more difficult. To solve this issue, we propose a scale-aware dimension-wise attention network (SA-DWA-Net) for better small ship instance segmentation in SAR images. SA-DWA-Net has two subnetworks to ensure its desirable instance segmentation of small ships. The first is a scale-aware subnetwork that can fully use low-level location-sensitive information to achieve representative small ship features. The second is a dimension-wise attention subnetwork that can fully utilize high-level semantics-sensitive information for refined small ship feature expression. We perform experiments on two open SSDD and HRSID datasets to verify the effectiveness of the proposed method. Quantitative experimental results show the state-of-the-art SAR ship instance segmentation performance of the proposed SA-DWA-Net. Specifically, SA-DWA-Net surpasses the existing best model by 2.2% box detection average precision (AP) and 5.0% mask segmentation AP on SSDD and by 2.9% box detection AP and 3.7% mask segmentation AP on HRSID. Especially, the small ship mask segmentation AP of the proposed SA-DWA-Net is higher than the existing best model by 4.4% on SSDD and 3.7% on HRSID.