Radar sensors play an essential role for robust automated driving especially under harsh weather conditions. As more and more radar sensors with increased numbers of transmitter and receiver channels are integrated into cars, the compute load for radar signal processing increases, also because AI models are increasingly used. To counteract the associated rising energy demand, neuromorphic hardware that realizes brain-inspired spiking neural networks in silicon may be used for energy-efficient radar signal processing. Here, we show a case study from the German KI-ASIC project evaluating the use of the SpiNNaker2 neuromorphic chip for automotive radar processing. First, we implement three different variants of the CA-CFAR algorithm (in software, as convolutional network, as spiking neural network) on the SpiNNaker2 chip and compare them regarding latency, energy and memory usage. Second, we propose and implement a pipeline for radar object tracking and classification on a test car with radar sensor, neuromorphic chip and conventional computer as middleware. This work presents by far the most elaborated approach for automotive radar processing with neuromorphic hardware and provides useful insights for future work.