We use spectra observed with the integral-field spectrograph MUSE to reveal the central kinematics of the Galactic globular cluster Messier 80 (M80, NGC 6093). Using observations obtained with the recently commissioned narrow-field mode of MUSE, we are able to analyse 932 stars in the central 7.5 arcsec by 7.5 arcsec of the cluster for which no useful spectra previously existed. Mean radial velocities of individual stars derived from the spectra are compared to predictions from axisymmetric Jeans models, resulting in radial profiles of the velocity dispersion, the rotation amplitude, and the mass-to-light ratio. The new data allow us to search for an intermediate-mass black hole (IMBH) in the centre of the cluster. Our Jeans model finds two similarly probable solutions around different dynamical cluster centres. The first solution has a centre close to the photometric estimates available in the literature and does not need an IMBH to fit the observed kinematics. The second solution contains a location of the cluster centre that is offset by about 2.4 arcsec from the first one and it needs an IMBH mass of $4600^{+1700}_{-1400}$ solar masses. N-body models support the existence of an IMBH in this cluster with a mass of up to 6000 solar masses in this cluster, although models without an IMBH provide a better fit to the observed surface brightness profile. They further indicate that the cluster has lost nearly all stellar-mass black holes. We further discuss the detection of two potential high-velocity stars with radial velocities of 80 to 90 km/s relative to the cluster mean.
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