In-car activity monitoring is a key enabler of various automotive safety functions. Existing approaches are largely based on vision systems. Radar, however, can provide a low-cost, privacy-preserving alternative. To this day, such systems based on the radar are not widely researched. In our work, we introduce a novel approach that uses the Doppler signal of an ultra-wideband (UWB) radar as an input to deep neural networks for the classification of driving activities. In contrast to previous work in the domain, we focus on generalization to unseen persons and make a new radar driving activity dataset (RaDA) available to the scientific community to encourage comparison and the benchmarking of future methods.