An ensemble of emitters can behave significantly different from its individual constituents when interacting coherently via a common light field. After excitation, collective coupling gives rise to an intriguing many-body quantum phenomenon, resulting in short, intense bursts of light: so-called superfluorescence. Because it requires a fine balance of interaction between the emitters and their decoupling from the environment, together with close identity of the individual emitters, superfluorescence has thus far been observed only in a limited number of systems, such as atomic and molecular gases and semiconductor crystals, and could not be harnessed for applications. For colloidal nanocrystals, however, which are of increasing relevance in a number of opto-electronic applications, the generation of superfluorescent light was precluded by inhomogeneous emission broadening, low oscillator strength, and fast exciton dephasing. Using caesium lead halide (CsPbX3, X = Cl, Br) perovskite nanocrystals that are self-organized into highly ordered three-dimensional superlattices allows us to observe key signatures of superfluorescence: red-shifted emission with more than ten-fold accelerated radiative decay, extension of the first-order coherence time by more than a factor of four, photon bunching, and delayed emission pulses with Burnham-Chiao ringing behaviour at high excitation density. These mesoscopically extended coherent states can be employed to boost opto-electronic device performances and enable entangled multi-photon quantum light sources.