Over the years, the use of superpixel segmentation has become very popular in various applications, serving as a preprocessing step to reduce data size by adapting to the content of the image, regardless of its semantic content. While the superpixel segmentation of standard planar images, captured with a 90° field of view, has been extensively studied, there has been limited focus on dedicated methods to omnidirectional or spherical images, captured with a 360° field of view. In this study, we introduce the first deep learning-based superpixel segmentation approach tailored for omnidirectional images called DSS (for Deep Spherical Superpixels). Our methodology leverages on spherical CNN architectures and the differentiable K-means clustering paradigm for superpixels, to generate superpixels that follow the spherical geometry. Additionally, we propose to use data augmentation techniques specifically designed for 360° images, enabling our model to efficiently learn from a limited set of annotated omnidirectional data. Our extensive validation across two datasets demonstrates that taking into account the inherent circular geometry of such images into our framework improves the segmentation performance over traditional and deep learning-based superpixel methods. Our code is available online.