Bats (Chiroptera) represent one of the largest and most diverse radiations of mammals, accounting for one-fifth of extant species. Although recent studies unambiguously support bat monophyly and consensus is rapidly emerging about evolutionary relationships among extant lineages, the fossil record of bats extends over 50 million years, and early evolution of the group remains poorly understood. Here we describe a new bat from the Early Eocene Green River Formation of Wyoming, USA, with features that are more primitive than seen in any previously known bat. The evolutionary pathways that led to flapping flight and echolocation in bats have been in dispute, and until now fossils have been of limited use in documenting transitions involved in this marked change in lifestyle. Phylogenetically informed comparisons of the new taxon with other bats and non-flying mammals reveal that critical morphological and functional changes evolved incrementally. Forelimb anatomy indicates that the new bat was capable of powered flight like other Eocene bats, but ear morphology suggests that it lacked their echolocation abilities, supporting a `flight first' hypothesis for chiropteran evolution. The shape of the wings suggests that an undulating gliding-fluttering flight style may be primitive for bats, and the presence of a long calcar indicates that a broad tail membrane evolved early in Chiroptera, probably functioning as an additional airfoil rather than as a prey-capture device. Limb proportions and retention of claws on all digits indicate that the new bat may have been an agile climber that employed quadrupedal locomotion and under-branch hanging behaviour.