Millimeter-wave (mmWave) technology is increasingly recognized as a pivotal technology of the sixth-generation communication networks due to the large amounts of available spectrum at high frequencies. However, the huge overhead associated with beam training imposes a significant challenge in mmWave communications, particularly in urban environments with high background noise. To reduce this high overhead, we propose a novel solution for robust continuous-time beam tracking with liquid neural network, which dynamically adjust the narrow mmWave beams to ensure real-time beam alignment with mobile users. Through extensive simulations, we validate the effectiveness of our proposed method and demonstrate its superiority over existing state-of-the-art deep-learning-based approaches. Specifically, our scheme achieves at most 46.9% higher normalized spectral efficiency than the baselines when the user is moving at 5 m/s, demonstrating the potential of liquid neural networks to enhance mmWave mobile communication performance.