Aims. Our goal is to probe the inner structure and the nature of the mid-IR emission in the active galaxy IC 3639, which hosts a Seyfert 2 nucleus and shows signatures of strong star-forming activity. Methods. We use interferometric observations in the N-band with VLTI/MIDI to resolve the mid-IR emission of this nucleus. 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. A large fraction of the total mid-IR emission of IC 3639 is produced in the innermost 80 pc with only ∼ 1% of the total luminosity released in the UV/optical range. 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). The emission confined within the inner 80 pc is either dominated by a starburst component or by the AGN core. The brightness distribution could be reproduced by a single component associated to the AGN, although this scenario would imply a very extended dust distribution when compared to other nearby Seyfert galaxies detected with MIDI. The extended component could also be associated to polar dust emission, i.e. a dusty wind blown by the AGN. However, a mixed contribution dominated by the star formation component 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.