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Dicke superradiance describes the collective radiative decay of a fully inverted ensemble of two-level atoms. We experimentally investigate this effect for a chiral, i.e.,~direction-dependent light--matter coupling. Despite a fundamentally different interaction Hamiltonian which has a reduced symmetry compared to the standard Dicke case, we do observe a superradiant burst emission. The burst occurs above a threshold number of atoms, and its peak power scales faster with the number of atoms than in the case of free-space Dicke superradiance. We measure the first-order coherence of the burst emission and experimentally distinguish two regimes, one dominated by the coherence induced during the excitation process and the other governed by vacuum fluctuations. Our results shed light on the collective radiative dynamics of cascaded quantum many-body systems, i.e., a system in which each quantum emitter is only driven by light radiated by emitters that are further upstream in the cascade. Our findings may turn out useful for generating multi-photon Fock states as a resource for quantum technologies.