Remote sensing of rainfall events is critical for both operational and scientific needs, including for example weather forecasting, extreme flood mitigation, water cycle monitoring, etc. Ground-based weather radars, such as NOAA's Next-Generation Radar (NEXRAD), provide reflectivity and precipitation measurements of rainfall events. However, the observation range of such radars is limited to a few hundred kilometers, prompting the exploration of other remote sensing methods, paricularly over the open ocean, that represents large areas not covered by land-based radars. For a number of decades, C-band SAR imagery such a such as Sentinel-1 imagery has been known to exhibit rainfall signatures over the sea surface. However, the development of SAR-derived rainfall products remains a challenge. Here we propose a deep learning approach to extract rainfall information from SAR imagery. We demonstrate that a convolutional neural network, such as U-Net, trained on a colocated and preprocessed Sentinel-1/NEXRAD dataset clearly outperforms state-of-the-art filtering schemes. Our results indicate high performance in segmenting precipitation regimes, delineated by thresholds at 1, 3, and 10 mm/h. Compared to current methods that rely on Koch filters to draw binary rainfall maps, these multi-threshold learning-based models can provide rainfall estimation for higher wind speeds and thus may be of great interest for data assimilation weather forecasting or for improving the qualification of SAR-derived wind field data.