The outskirts of globular clusters (GCs) simultaneously retain crucial information about their formation mechanism and the properties of their host galaxy. Thanks to the advent of precision astrometry both their morphological and kinematic properties are now accessible. Here we present the first dynamical study of the outskirts of the retrograde GC NGC 3201 until twice its Jacobi radius (< 100 pc), using specifically-selected high-quality astrometric data from Gaia DR2. We report the discovery of a stellar overdensity along the South-East/North-West direction that we identify as tidal tails. The GC is characterized globally by radial anisotropy and a hint of isotropy in the outer parts, with an excess of tangential orbits around the lobes corresponding to the tidal tails, in qualitative agreement with an N-body simulation. Moreover, we measure flat velocity dispersion profiles, reaching values of $3.5pm0.9$ km/s until beyond the Jacobi radius. While tidal tails could contribute to such a flattening, this high velocity dispersion value is in disagreement with the expectation from the sole presence of potential escapers. To explain this puzzling observation, we discuss the possibility of an accreted origin of the GC, the presence of a dark matter halo --leftover of its formation at high redshift -- and the possible effects of non-Newtonian dynamics. Our study uncovers a new path for the study of GC formation and of the properties of the Milky Way potential in the era of precision astrometry.
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