Aims. In this study we use multi-epoch near-infrared observations from the VISTA survey of the Magellanic Cloud system (VMC) to measure the proper motions of different stellar populations in a tile of 1.5 deg2 in size in the direction of the Galactic globular cluster 47 Tuc. We obtain the proper motion of the cluster itself, of the Small Magellanic Cloud (SMC), and of the field Milky Way stars. Methods. Stars of the three main stellar components are selected according to their spatial distributions and their distributions in colour-magnitude diagrams. Their average coordinate displacement is computed from the difference between multiple Ks-band observations for stars as faint as Ks = 19 mag. Proper motions are derived from the slope of the best-fitting line among ten VMC epochs over a time baseline of ~1 yr. Background galaxies are used to calibrate the absolute astrometric reference frame. Results. The resulting absolute proper motion of 47 Tuc is (μαcos(δ), μδ) = (+7.26 ± 0.03, -1.25 ± 0.03) mas yr-1. This measurement refers to about 35 000 sources distributed between 10′ and 60′ from the cluster centre. For the SMC we obtain (μαcos(δ), μδ) = (+1.16 ± 0.07, -0.81 ± 0.07) mas yr-1 from about 5250 red clump and red giant branch stars. The absolute proper motion of the Milky Way population in the line of sight (l = 305.9, b = -44.9) of this VISTA tile is (μαcos(δ), μδ) = (+10.22 ± 0.14, -1.27 ± 0.12) mas yr-1 and has been calculated from about 4000 sources. Systematic uncertainties associated with the astrometric reference system are 0.18 mas yr-1. Thanks to the proper motion we detect 47 Tuc stars beyond its tidal radius.