Context. On the basis of its low luminosity, its chemical composition, and the absence of a large-scale outflow, the dense core Cha-MMS1 located in the Chamaeleon I molecular cloud, was proposed a decade ago as a candidate for a first hydrostatic core (FHSC).Aims. Our goal is to test this hypothesis by searching for a slow, compact outflow driven by Cha-MMS1 that would match the predictions of magnetohydrodynamic simulations for this short phase of star formation.Methods. We used the Atacama Large Millimetre/submillimetre Array to map Cha-MMS1 at high angular resolution in CO 3–2 and 13CO 3–2 as well as in continuum emission.Results. We report the detection of a bipolar outflow emanating from the central core, along a (projected) direction roughly parallel to the filament in which Cha-MMS1 is embedded and perpendicular to the large-scale magnetic field. The morphology of the outflow indicates that its axis lies close to the plane of the sky. We measure velocities corrected for inclination of more than 90 km s−1, which is clearly incompatible with the expected properties of an FHSC outflow. Several properties of the outflow are determined and compared to previous studies of Class 0 and Class I protostars. The outflow of Cha-MMS1 has a much smaller momentum force than the outflows of other Class 0 protostars. In addition, we find a dynamical age of 200–3000 yr indicating that Cha-MMS1 might be one of the youngest ever observed Class 0 protostars. While the existence of the outflow suggests the presence of a disk, no disk is detected in continuum emission and we derive an upper limit of 55 au to its radius.Conclusions. We conclude that Cha-MMS1 has already gone through the FHSC phase and is a young Class 0 protostar, but it has not yet brought its outflow to full power.