Band selection refers to the process of choosing the most relevant bands in a hyperspectral image. By selecting a limited number of optimal bands, we aim at speeding up model training, improving accuracy, or both. It reduces redundancy among spectral bands while trying to preserve the original information of the image. By now many efforts have been made to develop unsupervised band selection approaches, of which the majority are heuristic algorithms devised by trial and error. In this paper, we are interested in training an intelligent agent that, given a hyperspectral image, is capable of automatically learning policy to select an optimal band subset without any hand-engineered reasoning. To this end, we frame the problem of unsupervised band selection as a Markov decision process, propose an effective method to parameterize it, and finally solve the problem by deep reinforcement learning. Once the agent is trained, it learns a band-selection policy that guides the agent to sequentially select bands by fully exploiting the hyperspectral image and previously picked bands. Furthermore, we propose two different reward schemes for the environment simulation of deep reinforcement learning and compare them in experiments. This, to the best of our knowledge, is the first study that explores a deep reinforcement learning model for hyperspectral image analysis, thus opening a new door for future research and showcasing the great potential of deep reinforcement learning in remote sensing applications. Extensive experiments are carried out on four hyperspectral data sets, and experimental results demonstrate the effectiveness of the proposed method.