To better resolve the bed elevation of Antarctica, we present DeepBedMap - a deep learning method that produces realistic Antarctic bed topography from multiple remote sensing data inputs. Our super-resolution deep convolutional neural network model is trained on scattered regions in Antarctica where high resolution (250 m) groundtruth bed elevation grids are available, and then used to generate high resolution bed topography in less well surveyed areas. DeepBedMap takes in a low resolution (1000 m) BEDMAP2 dataset alongside other high spatial resolution inputs such as ice surface elevation, velocity and snow accumulation to generate a four times upsampled (250 m) bed topography map even in the absence of ice-thickness data from direct seismic or ice-penetrating radar surveys. Our DeepBedMap model is based on an Enhanced Super Resolution Generative Adversarial Network architecture that is adapted to minimize per-pixel elevation errors while producing realistic topography. We show that DeepBedMap offers a more realistic topographic roughness profile compared to a standard bicubic interpolated BEDMAP2, and also run model inversions to compare the basal traction of our DeepBedMap_DEM with other bed elevation models.