Generalizable face anti-spoofing is a challenging task due to the variations of fake materials (e.g., paper, plastic, and silicon), attack types (e.g., physical and digital), and acquisition environment (e.g., lighting). In this paper, we propose a novel Hyperbolic Metric Learning method for generalizable Face Anti-Spoofing, namely HML-FAS. Compared with the widely used Euclidean metric learning, the inherent hierarchical structure of anti-spoofing data can be well captured in the hyperbolic metric space. In particular, HML-FAS consists of an initial hyperbolic feature embedding step, followed by a Hyperbolic adversarial Data Augmentation (HDA), a Hyperbolic Optimal Transport (HOT), and a final hyperbolic classifier. To learn robust features, the hyperbolic Stein variational gradient descent algorithm is used for HDA to broaden the feature distribution bounds of each training domain. To learn domain-invariant features, the Kantorovich potential network is utilized for HOT to map the feature distributions of all training domains to a common hyperbolic space. Combined with the final hyperbolic classifier, out-of-distribution robust, domain-invariant, and discriminative face anti-spoofing features can be learned by our HML-FAS. Extensive experiments and visualizations demonstrate the effectiveness of HML-FAS compared with its Euclidean version EML-FAS, and the previous state-of-the-art methods under unseen scenarios and for unknown attacks.