Advancements in face recognition (FR) technologies have amplified privacy concerns, necessitating methods that protect identity while maintaining recognition utility. Existing face anonymization methods typically focus on obscuring identity but fail to meet the requirements of biometric template protection, including revocability, unlinkability, and irreversibility. We propose FaceAnonyMixer, a cancelable face generation framework that leverages the latent space of a pre-trained generative model to synthesize privacy-preserving face images. The core idea of FaceAnonyMixer is to irreversibly mix the latent code of a real face image with a synthetic code derived from a revocable key. The mixed latent code is further refined through a carefully designed multi-objective loss to satisfy all cancelable biometric requirements. FaceAnonyMixer is capable of generating high-quality cancelable faces that can be directly matched using existing FR systems without requiring any modifications. Extensive experiments on benchmark datasets demonstrate that FaceAnonyMixer delivers superior recognition accuracy while providing significantly stronger privacy protection, achieving over an 11% gain on commercial API compared to recent cancelable biometric methods. Code is available at: https://github.com/talha-alam/faceanonymixer.