LiDAR semantic segmentation is important for understanding the surrounding environment in autonomous driving. Existing methods assume closed-set situations with the same training and testing label space. However, in the real world, unknown classes not encountered during training may appear during testing, making it difficult to apply existing methodologies. In this paper, we propose a novel on-the-fly category discovery method for LiDAR semantic segmentation, aiming to classify and segment both unknown and known classes instantaneously during test time, achieved solely by learning with known classes in training. To embed instant segmentation capability in an inductive setting, we adopt a hash coding-based model with an expandable prediction space as a baseline. Based on this, dual prototypical learning is proposed to enhance the recognition of the known classes by reducing the sensitivity to intra-class variance. Additionally, we propose a novel mixing-based category learning framework based on representation mixing to improve the discovery capability of unknown classes. The proposed mixing-based framework effectively models out-of-distribution representations and learns to semantically group them during training, while distinguishing them from in-distribution representations. Extensive experiments on SemanticKITTI and SemanticPOSS datasets demonstrate the superiority of the proposed methods compared to the baselines. The code will be released.