Purpose Hyperspectral imaging has the potential to improve intraoperative decision making if tissue characterisation is performed in real-time and with high-resolution. Hyperspectral snapshot mosaic sensors offer a promising approach due to their fast acquisition speed and compact size. However, a demosaicking algorithm is required to fully recover the spatial and spectral information of the snapshot images. Most state-of-the-art demosaicking algorithms require ground-truth training data with paired snapshot and high-resolution hyperspectral images, but such imagery pairs with the exact same scene are physically impossible to acquire in intraoperative settings. In this work, we present a fully unsupervised hyperspectral image demosaicking algorithm which only requires exemplar snapshot images for training purposes. Methods We regard hyperspectral demosaicking as an ill-posed linear inverse problem which we solve using a deep neural network. We take advantage of the spectral correlation occurring in natural scenes to design a novel inter spectral band regularisation term based on spatial gradient consistency. By combining our proposed term with standard regularisation techniques and exploiting a standard data fidelity term, we obtain an unsupervised loss function for training deep neural networks, which allows us to achieve real-time hyperspectral image demosaicking. Results Quantitative results on hyperspetral image datasets show that our unsupervised demosaicking approach can achieve similar performance to its supervised counter-part, and significantly outperform linear demosaicking. A qualitative user study on real snapshot hyperspectral surgical images confirms the results from the quantitative analysis. Conclusion Our results suggest that the proposed unsupervised algorithm can achieve promising hyperspectral demosaicking in real-time thus advancing the suitability of the modality for intraoperative use.