Millimeter-wave (mmWave) multiple-input multiple-output (MIMO) communication with the advanced beamforming technologies is a key enabler to meet the growing demands of future mobile communication. However, the dynamic nature of cellular channels in large-scale urban mmWave MIMO communication scenarios brings substantial challenges, particularly in terms of complexity and robustness. To address these issues, we propose a robust gradient-based liquid neural network (GLNN) framework that utilizes ordinary differential equation-based liquid neurons to solve the beamforming problem. Specifically, our proposed GLNN framework takes gradients of the optimization objective function as inputs to extract the high-order channel feature information, and then introduces a residual connection to mitigate the training burden. Furthermore, we use the manifold learning technique to compress the search space of the beamforming problem. These designs enable the GLNN to effectively maintain low complexity while ensuring strong robustness to noisy and highly dynamic channels. Extensive simulation results demonstrate that the GLNN can achieve 4.15% higher spectral efficiency than that of typical iterative algorithms, and reduce the time consumption to only 1.61% that of conventional methods.