We propose the concept of a spontaneous many-body Floquet state. This is a state that, in the absence of external periodic driving, still self-oscillates like in the presence of a Floquet Hamiltonian, this behavior being spontaneously induced by many-body interactions. Furthermore, we prove that it is also a time crystal, presenting long-range time-periodic order. However, its time crystalline behavior is very different to that of conventional Floquet discrete time crystals: here, there is no external periodic driving, and the nature of the spontaneous symmetry breaking is continuous instead of discrete. We also demonstrate that a spontaneous many-body Floquet state can be implemented in a one-dimensional flowing atom condensate, resulting from a dynamical phase transition and stable against quantum fluctuations, and propose realistic experimental scenarios for its observation. The realization of a spontaneous many-body Floquet state would then not only provide a novel form of ordered quantum matter, but also a continuous time crystal.