Automotive radar systems face the challenge of managing high sampling rates and large data bandwidth while complying with stringent real-time and energy efficiency requirements. The growing complexity of autonomous vehicles further intensifies these requirements. Neuromorphic computing offers promising solutions because of its inherent energy efficiency and parallel processing capacity. This research presents a novel spiking neuron model for signal processing of frequency-modulated continuous wave (FMCW) radars that outperforms the state-of-the-art spectrum analysis algorithms in latency and data bandwidth. These spiking neural resonators are based on the resonate-and-fire neuron model and optimized to dynamically process raw radar data while simultaneously emitting an output in the form of spikes. We designed the first neuromorphic neural network consisting of these spiking neural resonators that estimates range and angle from FMCW radar data. We evaluated the range-angle maps on simulated datasets covering multiple scenarios and compared the results with a state-of-the-art pipeline for radar processing. The proposed neuron model significantly reduces the processing latency compared to traditional frequency analysis algorithms, such as the Fourier transformation (FT), which needs to sample and store entire data frames before processing. The evaluations demonstrate that these spiking neural resonators achieve state-of-the-art detection accuracy while emitting spikes simultaneously to processing and transmitting only 0.02 % of the data compared to a float-32 FT. The results showcase the potential for neuromorphic signal processing for FMCW radar systems and pave the way for designing neuromorphic radar sensors.