The Greenland ice sheet is melting at unprecedented rates. In this regard, it is crucial to better understand the processes that lead to melting and sea level rise. A common indicator for these processes is surface meltwater, which is water that forms on top of or within the first meter of the ice sheet. The highest resolution information on surface meltwater can be derived from satellites with a synthetic aperture radar (SAR) instrument, but the resulting data is hard to use due to temporal gaps from the satellite's revisit period. During such temporal gaps an extreme meltwater event that can produce billions of tons of meltwater within a single day could have occurred. To simplify the use of surface meltwater data, we propose a deep learning method that creates regularly-spaced, daily, high-resolution maps of surface meltwater. The deep learning model does so by fusing the information from SAR with other satellite data and physics-based simulations that are available on a daily basis. We show that surface meltwater maps from our deep learning model are significantly more accurate than currently used maps for an important glacier in Greenland. And, we publish our data as benchmark to encourage development of future models.