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Frequency-modulated continuous wave radar sensors play an essential role for assisted and autonomous driving as they are robust under all weather and light conditions. However, the rising number of transmitters and receivers for obtaining a higher angular resolution increases the cost for digital signal processing. One promising approach for energy-efficient signal processing is the usage of brain-inspired spiking neural networks (SNNs) implemented on neuromorphic hardware. In this article we perform a step-by-step analysis of automotive radar processing and argue how spiking neural networks could replace or complement the conventional processing. We provide SNN examples for two processing steps and evaluate their accuracy and computational efficiency. For radar target detection, an SNN with temporal coding is competitive to the conventional approach at a low compute overhead. Instead, our SNN for target classification achieves an accuracy close to a reference artificial neural network while requiring 200 times less operations. Finally, we discuss the specific requirements and challenges for SNN-based radar processing on neuromorphic hardware. This study proves the general applicability of SNNs for automotive radar processing and sustains the prospect of energy-efficient realizations in automated vehicles.