This paper presents a method to learn the Cartesian velocity of objects using an object detection network on automotive radar data. The proposed method is self-supervised in terms of generating its own training signal for the velocities. Labels are only required for single-frame, oriented bounding boxes (OBBs). Labels for the Cartesian velocities or contiguous sequences, which are expensive to obtain, are not required. The general idea is to pre-train an object detection network without velocities using single-frame OBB labels, and then exploit the network's OBB predictions on unlabelled data for velocity training. In detail, the network's OBB predictions of the unlabelled frames are updated to the timestamp of a labelled frame using the predicted velocities and the distances between the updated OBBs of the unlabelled frame and the OBB predictions of the labelled frame are used to generate a self-supervised training signal for the velocities. The detection network architecture is extended by a module to account for the temporal relation of multiple scans and a module to represent the radars' radial velocity measurements explicitly. A two-step approach of first training only OBB detection, followed by training OBB detection and velocities is used. Further, a pre-training with pseudo-labels generated from radar radial velocity measurements bootstraps the self-supervised method of this paper. Experiments on the publicly available nuScenes dataset show that the proposed method almost reaches the velocity estimation performance of a fully supervised training, but does not require expensive velocity labels. Furthermore, we outperform a baseline method which uses only radial velocity measurements as labels.