Speech enhancement (SE) is crucial for reliable communication devices or robust speech recognition systems. Although conventional artificial neural networks (ANN) have demonstrated remarkable performance in SE, they require significant computational power, along with high energy costs. In this paper, we propose a novel approach to SE using a spiking neural network (SNN) based on a U-Net architecture. SNNs are suitable for processing data with a temporal dimension, such as speech, and are known for their energy-efficient implementation on neuromorphic hardware. As such, SNNs are thus interesting candidates for real-time applications on devices with limited resources. The primary objective of the current work is to develop an SNN-based model with comparable performance to a state-of-the-art ANN model for SE. We train a deep SNN using surrogate-gradient-based optimization and evaluate its performance using perceptual objective tests under different signal-to-noise ratios and real-world noise conditions. Our results demonstrate that the proposed energy-efficient SNN model outperforms the Intel Neuromorphic Deep Noise Suppression Challenge (Intel N-DNS Challenge) baseline solution and achieves acceptable performance compared to an equivalent ANN model.