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The broad usage of mobile devices nowadays, the sensitiveness of the information contained in them, and the shortcomings of current mobile user authentication methods are calling for novel, secure, and unobtrusive solutions to verify the users' identity. In this article, we propose TypeFormer, a novel Transformer architecture to model free-text keystroke dynamics performed on mobile devices for the purpose of user authentication. The proposed model consists in Temporal and Channel Modules enclosing two Long Short-Term Memory (LSTM) recurrent layers, Gaussian Range Encoding (GRE), a multi-head Self-Attention mechanism, and a Block-Recurrent structure. Experimenting on one of the largest public databases to date, the Aalto mobile keystroke database, TypeFormer outperforms current state-of-the-art systems achieving Equal Error Rate (EER) values of 3.25% using only 5 enrolment sessions of 50 keystrokes each. In such way, we contribute to reducing the traditional performance gap of the challenging mobile free-text scenario with respect to its desktop and fixed-text counterparts. Additionally, we analyse the behaviour of the model with different experimental configurations such as the length of the keystroke sequences and the amount of enrolment sessions, showing margin for improvement with more enrolment data. Finally, a cross-database evaluation is carried out, demonstrating the robustness of the features extracted by TypeFormer in comparison with existing approaches.