Context. Proper motion (PM) studies are fundamental ingredients in the understanding of the orbital history of galaxies. Current measurements do not yet provide a satisfactory answer to the possible scenarios for the formation and evolution of the Magellanic Clouds and of the Bridge and Stream that link them with each other and with our Galaxy. Aims. We use multi-epoch near-infrared observations from the VISTA survey of the Magellanic Cloud system (VMC) to measure the PM of stars of the Large Magellanic Cloud (LMC), in one tile of 1.5 deg2 centred at (α,δ) = (05:59:23.136, -66:20:28.68) and including the south ecliptic pole, with respect to their Two Micron All Sky Survey (2MASS) position over a time baseline of about 10 years. Proper motions from VMC observations only, spanning a time range of about 1 year, are also derived. Methods. Stars of different ages are selected from the colour-magnitude diagram, (J - Ks) vs. Ks, and their average coordinate displacement is computed from the difference between Ks band observations from VMC and 2MASS or among VMC data alone for stars as faint as Ks = 19 mag. Proper motions are derived by averaging up to seven 2MASS-VMC combinations in the first case and from the slope of the best-fit line among the seven VMC epochs in the second case. Separate PM values are obtained for Cepheids, RR Lyrae stars, long period variables, and eclipsing binary stars in the field. Results. The PM of ~40 000 LMC stars in the tile, with respect to ~8000 background galaxies, obtained from VMC data alone, is μ αcos (δ) = +2.20 ± 0.06 (stat) ±0.29 (sys) and μδ = +1.70 ± 0.06 (stat) ±0.30 (sys) mas yr-1. This value agrees with recent ground-based determinations, but is larger than studies with the Hubble Space Telescope; this discrepancy may be due to additional systematic errors in the data. Our result implies either higher tangential motion or higher internal motion, or a combination of these, although we cannot discuss these possibilities quantitatively based on one field. The PM of the LMC is also clearly distinct from the PM derived for stars in the Milky Way foreground. The relative PM between the foreground stars and the LMC stars is ~5 mas yr-1. Furthermore, we measure a decrease in the PM with increasing logarithm of stellar age for LMC stars. Conclusions. This study, based on just one VMC tile, shows the potential of the 2MASS-VMC and VMC-VMC combinations for a comprehensive investigation of PM across the Magellanic system. © ESO, 2014.