Reliable information about the spatial distribution of surface waters is critically important in various scientific disciplines. Synthetic Aperture Radar (SAR) is an effective way to detect floods and monitor water bodies over large areas. Sentinel-1 is a new available SAR and its spatial resolution and short temporal baselines have the potential to facilitate the monitoring of surface water changes, which are dynamic in space and time. While several methods and tools for flood detection and surface water extraction already exist, they often comprise a significant manual user interaction and do not specifically target the exploitation of Sentinel-1 data. The existing methods commonly rely on thresholding at the level of individual pixels, ignoring the correlation among nearby pixels. Thus, in this paper, we propose a fully automatic processing chain for rapid flood and surface water mapping with smooth labeling based on Sentinel-1 amplitude data. The method is applied to three different sites submitted to recent flooding events. The quantitative evaluation shows relevant results with overall accuracies of more than 98% and F-measure values ranging from 0.64 to 0.92. These results are encouraging and the first step to proposing operational image chain processing to help end-users quickly map flooding events or surface waters.