Context. First hydrostatic cores represent a theoretically predicted intermediate evolutionary link between the prestellar and protostellar phases. Studying the observational characteristics of first core candidates is therefore vital for probing and understanding the earliest phases of star formation. Aims. We aim to determine the dynamical state of the first hydrostatic core candidate Chamaeleon-MMS1 (Cha-MMS1). Methods. We observed Cha-MMS1 in various molecular transitions with the APEX and Mopra telescopes. Continuum data retrieved from the Spitzer Heritage Archive were used to estimate the internal luminosity of the source. The molecular emission was modelled with a radiative transfer code to derive constraints on the kinematics of the envelope, which were then compared to the predictions of magneto-hydrodynamic simulations. Results. We derive an internal luminosity of 0.08 L −0.18 L for Cha-MMS1. An average velocity gradient of 3.1 ± 0.1 km s −1 pc −1 over ∼0.08 pc is found perpendicular to the filament in which Cha-MMS1 is embedded. The gradient is flatter in the outer parts and, surprisingly, also at the innermost ∼2000 AU to 4000 AU. The former features are consistent with solid-body rotation beyond 4000 AU and slower, differential rotation beyond 8000 AU, but the origin of the flatter gradient in the innermost parts is unclear. The classical infall signature is detected in HCO + 3−2 and CS 2−1. The radiative transfer modelling indicates a uniform infall velocity in the outer parts of the envelope. In the inner parts (at most 9000 AU), an infall velocity field scaling with r −0.5 is consistent with the data, but the shape of the profile is less well constrained and the velocity could also decrease toward the centre. The infall velocities are subsonic to transonic, 0.1 km s −1 −0.2 km s −1 at r ≥ 3300 AU, and subsonic to supersonic, 0.04 km s −1 −0.6 km s −1 at r ≤ 3300 AU. Both the internal luminosity of Cha-MMS1 and the infall velocity field in its envelope are consistent with predictions of MHD simulations for the first core phase. There is no evidence of any fast, large-scale outflow stemming from Cha-MMS1, but excess emission from the high-density tracers CS 5−4, CO 6−5, and CO 7−6 suggests the presence of higher velocity material at the inner core. Conclusions. Its internal luminosity excludes Cha-MMS1 being a prestellar core. The kinematical properties of its envelope are consistent with Cha-MMS1 being a first hydrostatic core candidate or a very young Class 0 protostar.