In the colliding-wind region of massive binaries, non-thermal radio emission occurs. This non-thermal radio emission (due to synchrotron radiation) has so far been observed at centimetre wavelengths. At millimetre wavelengths, the stellar winds and the colliding-wind region emit more thermal free-free radiation, and it is expected that any non-thermal contribution will be difficult or impossible to detect. We aim to determine if the material in the colliding-wind region contributes substantially to the observed millimetre fluxes of a colliding-wind binary. We also try to distinguish the synchrotron emission from the free-free emission. We monitored the massive binary Cyg OB2 #8A at 3 mm with the NOrthern Extended Millimeter Array (NOEMA) interferometer of the Institut de Radioastronomie Millimetrique (IRAM). The data were collected in 14 separate observing runs (in 2014 and 2016), and provide good coverage of the orbital period. The observed millimetre fluxes range between 1.1 and 2.3 mJy, and show phase-locked variability, clearly indicating that a large part of the emission is due to the colliding-wind region. A simple synchrotron model gives fluxes with the correct order of magnitude, but with a maximum that is phase-shifted with respect to the observations. Qualitatively this phase shift can be explained by our neglect of orbital motion on the shape of the colliding-wind region. A model using only free-free emission results in only a slightly worse explanation of the observations. Additionally, on the map of our observations we also detect the O6.5 III star Cyg OB2 #8B, for which we determine a 3 mm flux of 0.21 +- 0.033 mJy. The question of whether synchrotron radiation or free-free emission dominates the millimetre fluxes of Cyg OB2 #8A remains open. More detailed modelling of this system, based on solving the hydrodynamical equations, is required to give a definite answer.
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