Vocal imitation--the ability to learn a previously unknown acoustic signal from a tutor--is considered to be a key innovation in the evolution of speech. This faculty is very rare and patchily distributed within the animal kingdom, suggesting multiple instances of convergent evolution. It has long been predicted that bats should be capable of vocal imitation and our results provide evidence for this phenomenon. We report that pups of the bat Saccopteryx bilineata learn a complex vocalization through vocal imitation. During ontogeny, pups of both sexes imitate territorial song from adult males, starting with simple precursor songs that develop into genuine renditions. The resemblance of pup renditions to their acoustic model is not caused by physical maturation effects, is independent of pups' gender and relatedness towards adult males and becomes more pronounced during ontogeny, showing that auditory experience is essential for vocal development. Our findings indicate that the faculty of vocal imitation is more widespread than previously thought and emphasize the importance of research on audiovocal communication in bats for a better understanding of the evolutionary origin of vocal imitation.