The field of neuromorphic computing has gained significant attention in recent years, aiming to bridge the gap between the efficiency of biological neural networks and the performance of artificial intelligence systems. This paper introduces Mamba-Spike, a novel neuromorphic architecture that integrates a spiking front-end with the Mamba backbone to achieve efficient and robust temporal data processing. The proposed approach leverages the event-driven nature of spiking neural networks (SNNs) to capture and process asynchronous, time-varying inputs, while harnessing the power of the Mamba backbone's selective state spaces and linear-time sequence modeling capabilities to model complex temporal dependencies effectively. The spiking front-end of Mamba-Spike employs biologically inspired neuron models, along with adaptive threshold and synaptic dynamics. These components enable efficient spatiotemporal feature extraction and encoding of the input data. The Mamba backbone, on the other hand, utilizes a hierarchical structure with gated recurrent units and attention mechanisms to capture long-term dependencies and selectively process relevant information. To evaluate the efficacy of the proposed architecture, a comprehensive empirical study is conducted on both neuromorphic datasets, including DVS Gesture and TIDIGITS, and standard datasets, such as Sequential MNIST and CIFAR10-DVS. The results demonstrate that Mamba-Spike consistently outperforms state-of-the-art baselines, achieving higher accuracy, lower latency, and improved energy efficiency. Moreover, the model exhibits robustness to various input perturbations and noise levels, highlighting its potential for real-world applications. The code will be available at https://github.com/ECNU-Cross-Innovation-Lab/Mamba-Spike.