Accurate estimates of surface solar irradiance (SSI) are essential for solar resource assessments and solar energy forecasts in grid integration and building control applications. SSI estimates for spatially extended regions can be retrieved from geostationary satellites such as Meteosat. Traditional SSI satellite retrievals like Heliosat rely on physical radiative transfer modelling. We introduce the first machine-learning-based satellite retrieval for instantaneous SSI and demonstrate its capability to provide accurate and generalizable SSI estimates across Europe. Our deep learning retrieval provides near real-time SSI estimates based on data-driven emulation of Heliosat and fine-tuning on pyranometer networks. By including SSI from ground stations, our SSI retrieval model can outperform Heliosat accuracy and generalize well to regions with other climates and surface albedos in cloudy conditions (clear-sky index < 0.8). We also show that the SSI retrieved from Heliosat exhibits large biases in mountain regions, and that training and fine-tuning our retrieval models on SSI data from ground stations strongly reduces these biases, outperforming Heliosat. Furthermore, we quantify the relative importance of the Meteosat channels and other predictor variables like solar zenith angle for the accuracy of our deep learning SSI retrieval model in different cloud conditions. We find that in cloudy conditions multiple near-infrared and infrared channels enhance the performance. Our results can facilitate the development of more accurate satellite retrieval models of surface solar irradiance.