As an important topic in hyperspectral image (HSI) analysis, band selection has attracted increasing attention in the last two decades for dimensionality reduction in HSI. With the great success of deep learning (DL)-based models recently, a robust unsupervised band selection (UBS) neural network is highly desired, particularly due to the lack of sufficient ground truth information to train the DL networks. Existing DL models for band selection either depend on the class label information or have unstable results via ranking the learned weights. To tackle these challenging issues, in this article, we propose a Gumbel-Softmax (GS) trick enabled concrete autoencoder-based UBS framework (CAE-UBS) for HSI, in which the learning process is featured by the introduced concrete random variables and the reconstruction loss. By searching from the generated potential band selection candidates from the concrete encoder, the optimal band subset can be selected based on an information entropy (IE) criterion. The idea of the CAE-UBS is quite straightforward, which does not rely on any complicated strategies or metrics. The robust performance on four publicly available datasets has validated the superiority of our CAE-UBS framework in the classification of the HSIs.