Recently, synthetic palmprints have been increasingly used as substitutes for real data to train recognition models. To be effective, such synthetic data must reflect the diversity of real palmprints, including both style variation and geometric variation. However, existing palmprint generation methods mainly focus on style translation, while geometric variation is either ignored or approximated by simple handcrafted augmentations. In this work, we propose FlowPalm, an optical-flow-driven palmprint generation framework capable of simulating the complex non-rigid deformations observed in real palms. Specifically, FlowPalm estimates optical flows between real palmprint pairs to capture the statistical patterns of geometric deformations. Building on these priors, we design a progressive sampling process that gradually introduces the geometric deformations during diffusion while maintaining identity consistency. Extensive experiments on six benchmark datasets demonstrate that FlowPalm significantly outperforms state-of-the-art palmprint generation approaches in downstream recognition tasks. Project page: https://yuchenzou.github.io/FlowPalm/