[Abridged] Methods: We use interferometric observations in the N-band with VLTI/MIDI to resolve the mid-IR nucleus of IC 3639. The origin of the nuclear infrared emission is determined from: 1) the comparison of the correlated fluxes from VLTI/MIDI with the fluxes measured at subarcsec resolution (VLT/VISIR, VLT/ISAAC); 2) diagnostics based on IR fine-structure line ratios, the IR continuum emission, IR bands produced by polycyclic aromatic hydrocarbons (PAH) and silicates; and 3) the high-angular resolution spectral energy distribution. Results: The unresolved flux of IC 3639 is 90 +/- 20 mJy at 10.5 µm, measured with three different baselines in VLTI (UT1-UT2, UT3-UT4, and UT2-UT3; 46-58 m), making this the faintest measurement so far achieved with mid-IR interferometry. The correlated flux is a factor of 3-4 times fainter than the VLT/VISIR total flux measurement. The observations suggest that most of the mid-IR emission has its origin on spatial scales between 10 and 80 pc (40-340 mas). A composite scenario where the star formation component dominates over the AGN is favoured by the diagnostics based on ratios of IR fine-structure emission lines, the shape of the IR continuum, and the PAH and silicate bands. Conclusions: A composite AGN-starburst scenario is able to explain both the mid-IR brightness distribution and the IR spectral properties observed in the nucleus of IC 3639. The nuclear starburst would dominate the mid-IR emission and the ionisation of low-excitation lines (e.g. [NeII]12.8µm) with a net contribution of ~70%. The AGN accounts for the remaining ~30% of the mid-IR flux, ascribed to the unresolved component in the MIDI observations, and the ionisation of high-excitation lines (e.g. [NeV]14.3µm and [OIV]25.9µm).