This work introduces and demonstrates the first system capable of imaging fast-moving extended non-rigid objects through strong atmospheric turbulence at high frame rate. Event cameras are a novel sensing architecture capable of estimating high-speed imagery at thousands of frames per second. However, on their own event cameras are unable to disambiguate scene motion from turbulence. In this work, we overcome this limitation using event-based light field cameras: By simultaneously capturing multiple views of a scene, event-based light field cameras and machine learning-based reconstruction algorithms are able to disambiguate motion-induced dynamics, which produce events that are strongly correlated across views, from turbulence-induced dynamics, which produce events that are weakly correlated across view. Tabletop experiments demonstrate event-based light field can overcome strong turbulence while imaging high-speed objects traveling at up to 16,000 pixels per second.