Gaining information about how users interact with systems is key to behavioural biometrics. Particularly mouse movements of users have been proven beneficial to authentication tasks for being inexpensive and non-intrusive. State-of-the-art approaches consider this problem an instance of supervised classification tasks. In this paper, we argue that the problem is actually closer to unsupervised one-class classification tasks. We thus propose to view behavioural user authentication as an unsupervised task and learn individual models using data from a single user only. We further show that, by being purely unsupervised, losses in performance can be counterbalanced by augmenting additional data into the training processes (outlier exposure). Empirical results show that our approach is very effective and outperforms the state-of-the-art in several performance metrics. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.