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Abstract The Dicke model—a paradigmatic example of superradiance in quantum optics—describes an ensemble of atoms which are collectively coupled to a leaky cavity mode. As a result of the cooperative nature of these interactions, the system’s dynamics is captured by the behavior of a single mean-field, collective spin. In this mean-field limit, it has recently been shown that the interplay between photon losses and periodic driving of light–matter coupling can lead to time-crystalline-like behavior of the collective spin (Gonget al2018Phys. Rev. Lett.120040404). In this work, we investigate whether such a Dicke time crystal (TC) is stable to perturbations that explicitly break the mean-field solvability of the conventional Dicke model. In particular, we consider the addition of short-range interactions between the atoms which breaks the collective coupling and leads to complex many-body dynamics. In this context, the interplay between periodic driving, dissipation and interactions yields a rich set of dynamical responses, including long-lived and metastable Dicke-TCs, where losses can cool down the many-body heating resulting from the continuous pump of energy from the periodic drive. Specifically, when the additional short-range interactions are ferromagnetic, we observe time crystalline behavior at non-perturbative values of the coupling strength, suggesting the possible existence of stable dynamical order in a driven-dissipative quantum many-body system. These findings illustrate the rich nature of novel dynamical responses with many-body character in quantum optics platforms.