An ensemble of emitters can behave very differently from its individual constituents when they interact coherently via a common light field. After excitation of such an ensemble, collective coupling can give rise to a many-body quantum phenomenon that results in short, intense bursts of light-so-called superfluorescence(1). Because this phenomenon requires a fine balance of interactions 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 certain atomic and molecular gases and a few solid-state systems(2-7). The generation of superfluorescent light in colloidal nanocrystals (which are bright photonic sources practically suited for optoelectronics(8,9)) has been precluded by inhomogeneous emission broadening, low oscillator strength, and fast exciton dephasing. Here we show that caesium lead halide (CsPbX3, X = Cl, Br) perovskite nanocrystals(10-13) that are self-organized into highly ordered three-dimensional superlattices exhibit key signatures of superfluorescence. These are dynamically red-shifted emission with more than 20-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(14) at high excitation density. These mesoscopically extended coherent states could be used to boost the performance of opto-electronic devices(15) and enable entangled multi-photon quantum light sources(16,17).