Event-based neural networks are currently being explored as efficient solutions for performing AI tasks at the extreme edge. To fully exploit their potential, event-based neural networks coupled to adequate preprocessing must be investigated. Within this context, we demonstrate a 4-b-weight spiking neural network (SNN) for radar gesture recognition, achieving a state-of-the-art 93% accuracy within only four processing time steps while using only one convolutional layer and two fully connected layers. This solution consumes very little energy and area if implemented in event-based hardware, which makes it suited for embedded extreme-edge applications. In addition, we demonstrate the importance of signal preprocessing for achieving this high recognition accuracy in SNNs compared to deep neural networks (DNNs) with the same network topology and training strategy. We show that efficient preprocessing prior to the neural network is drastically more important for SNNs compared to DNNs. We also demonstrate, for the first time, that the preprocessing parameters can affect SNNs and DNNs in antagonistic ways, prohibiting the generalization of conclusions drawn from DNN design to SNNs. We demonstrate our findings by comparing the gesture recognition accuracy achieved with our SNN to a DNN with the same architecture and similar training. Unlike previously proposed neural networks for radar processing, this work enables ultralow-power radar-based gesture recognition for extreme-edge devices.