Clustering of hyperspectral images is a fundamental but challenging task. The recent development of hyperspectral image clustering has evolved from shallow models to deep and achieved promising results in many benchmark datasets. However, their poor scalability, robustness, and generalization ability, mainly resulting from their offline clustering scenarios, greatly limit their application to large-scale hyperspectral data. To circumvent these problems, we present a scalable deep online clustering model, named Spectral-Spatial Contrastive Clustering (SSCC), based on self-supervised learning. Specifically, we exploit a symmetric twin neural network comprised of a projection head with a dimensionality of the cluster number to conduct dual contrastive learning from a spectral-spatial augmentation pool. We define the objective function by implicitly encouraging within-cluster similarity and reducing between-cluster redundancy. The resulting approach is trained in an end-to-end fashion by batch-wise optimization, making it robust in large-scale data and resulting in good generalization ability for unseen data. Extensive experiments on three hyperspectral image benchmarks demonstrate the effectiveness of our approach and show that we advance the state-of-the-art approaches by large margins.