Electroencephalography (EEG) is a non-invasive brain-computer interface technology used for recording brain electrical activity. It plays an important role in human life and has been widely uesd in real life, including sleep staging, emotion recognition, and motor imagery. However, existing EEG-related models cannot be well applied in practice, especially in clinical settings, where new patients with individual discrepancies appear every day. Such EEG-based model trained on fixed datasets cannot generalize well to the continual flow of numerous unseen subjects in real-world scenarios. This limitation can be addressed through continual learning (CL), wherein the CL model can continuously learn and advance over time. Inspired by CL, we introduce a novel Unsupervised Individual Continual Learning paradigm for handling this issue in practice. We propose the BrainUICL framework, which enables the EEG-based model to continuously adapt to the incoming new subjects. Simultaneously, BrainUICL helps the model absorb new knowledge during each adaptation, thereby advancing its generalization ability for all unseen subjects. The effectiveness of the proposed BrainUICL has been evaluated on three different mainstream EEG tasks. The BrainUICL can effectively balance both the plasticity and stability during CL, achieving better plasticity on new individuals and better stability across all the unseen individuals, which holds significance in a practical setting.