With the rapid growth of the Internet of Things ecosystem, Automatic Modulation Classification (AMC) has become increasingly paramount. However, extended signal lengths offer a bounty of information, yet impede the model's adaptability, introduce more noise interference, extend the training and inference time, and increase storage overhead. To bridge the gap between these requisites, we propose a novel AMC framework, designated as the Mamba-based Automatic Modulation ClassificAtion (MAMCA). Our method adeptly addresses the accuracy and efficiency requirements for long-sequence AMC. Specifically, we introduce the Selective State Space Model as the backbone, enhancing the model efficiency by reducing the dimensions of the state matrices and diminishing the frequency of information exchange across GPU memories. We design a denoising-capable unit to elevate the network's performance under low signal-to-noise radio. Rigorous experimental evaluations on the publicly available dataset RML2016.10, along with our synthetic dataset within multiple quadrature amplitude modulations and lengths, affirm that MAMCA delivers superior recognition accuracy while necessitating minimal computational time and memory occupancy. Codes are available on https://github.com/ZhangYezhuo/MAMCA.