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Employing deep neural networks for Hyperspectral remote sensing (HSRS) image classification is a challenging task. HSRS images have high dimensionality and a large number of channels with substantial redundancy between channels. In addition, the training data for classifying HSRS images is limited and the amount of available training data is much smaller compared to other classification tasks. These factors complicate the training process of deep neural networks with many parameters and cause them to not perform well even compared to conventional models. Moreover, convolutional neural networks produce over-confident predictions, which is highly undesirable considering the aforementioned problem. In this work, we use for HSRS image classification a special class of deep neural networks, namely a Bayesian neural network (BNN). To the extent of our knowledge, this is the first time that BNNs are used in HSRS image classification. BNNs inherently provide a measure for uncertainty. We perform extensive experiments on the Pavia Centre, Salinas, and Botswana datasets. We show that a BNN outperforms a standard convolutional neural network (CNN) and an off-the-shelf Random Forest (RF). Further experiments underline that the BNN is more stable and robust to model pruning, and that the uncertainty is higher for samples with higher expected prediction error.