  The formation of a phase of matter can be associated with the spontaneous
breaking of a symmetry. For crystallization, this broken symmetry is the
spatial translation symmetry, as the atoms spontaneously localize in a periodic
fashion. In analogy to spatial crystals, the spontaneous breaking of temporal
translation symmetry results in the formation of time crystals. While recent
and on-going experiments on driven isolated systems aim to minimize dissipative
processes, as it is an undesired source of decay, well-designed dissipation has
been put forth as a constitutive ingredient in the formation of dissipative
time crystals (DTCs). Here, we present the first experimental realisation of a
DTC, implemented in an atom-cavity system. Its defining feature is a period
doubled switching between distinct chequerboard density wave patterns, induced
by controlled cavity-dissipation and cavity-mediated interactions. We
demonstrate the robustness of this phase against system parameter changes and
temporal perturbations of the driving. Our work provides a framework for
realising phases of matter with spatiotemporal order in presence of
dissipation. We note that this is the natural environment of matter, and
therefore shapes its physical phenomena profoundly, making its study
imperative.
