Showing the record strengths and growth-rates, recent hurricanes have highlighted needs for improving forecasts of tropical cyclone intensities most sensitive to models of the air-sea interaction. Especially challenging is the nature of sea-spray supposed to strongly affecting the momentum- and energy- air-sea fluxes at strong winds. Even the spray-generation mechanisms in extreme winds remained undetermined. Basing on high-speed video here we identify it as the bag-breakup mode of fragmentation of liquid in gaseous flows known in a different context. This regime is characterized by inflating and consequent bursting of the short-lived objects, bags, comprising sail-like water films surrounded by massive liquid rims then fragmented to giant droplets with sizes exceeding 500 micrometers. From first principles of statistical physics we develop statistical description of these phenomena and show that at extreme winds the bag-breakup is the dominant spray-production mechanism. These findings provide a new basis for understanding and modeling of the air-sea exchange processes at extreme winds.