Hyperbolic neural networks (HNNs) are emerging as a promising tool for representing data embedded in non-Euclidean geometries, yet their adoption has been hindered by challenges related to stability and robustness. In this work, we conduct a rigorous Lipschitz analysis for HNNs and propose using Lipschitz regularization as a novel strategy to enhance their robustness. Our comprehensive investigation spans both the Poincaré ball model and the hyperboloid model, establishing Lipschitz bounds for HNN layers. Importantly, our analysis provides detailed insights into the behavior of the Lipschitz bounds as they relate to feature norms, particularly distinguishing between scenarios where features have unit norms and those with large norms. Further, we study regularization using the derived Lipschitz bounds. Our empirical validations demonstrate consistent improvements in HNN robustness against noisy perturbations.