We present the results of a spectroscopic survey performed in the outskirts of the globular cluster NGC 1851 with VIsible MultiObject Spectrograph (VIMOS)/Very Large Telescope (VLT). The radial velocities of 107 stars in a region between 12 and 33 arcmin around the cluster have been derived. We clearly identify the cluster stellar population over the entire field of view, indicating the presence of a significant fraction of stars outside the tidal radius predicted by King models. We also find tentative evidence of a cold (σv ≤ 20 km s−1) peak in the distribution of velocities at vr ∼ 180 km s−1 constituted mainly by main-sequence stars whose location in the colour–magnitude diagram is compatible with a stream at a similar distance to this cluster. If confirmed, this evidence would strongly support the extragalactic origin of this feature.