Flood mapping is crucial for assessing and mitigating flood impacts, yet traditional methods like numerical modeling and aerial photography face limitations in efficiency and reliability. To address these challenges, we propose PIFF, a physics-informed, flow-based generative neural network for near real-time flood depth estimation. Built on an image-to-image generative framework, it efficiently maps Digital Elevation Models (DEM) to flood depth predictions. The model is conditioned on a simplified inundation model (SPM) that embeds hydrodynamic priors into the training process. Additionally, a transformer-based rainfall encoder captures temporal dependencies in precipitation. Integrating physics-informed constraints with data-driven learning, PIFF captures the causal relationships between rainfall, topography, SPM, and flooding, replacing costly simulations with accurate, real-time flood maps. Using a 26 km study area in Tainan, Taiwan, with 182 rainfall scenarios ranging from 24 mm to 720 mm over 24 hours, our results demonstrate that PIFF offers an effective, data-driven alternative for flood prediction and response.