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 used 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 subarcsecond 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 with 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 with polar dust emission, that is, with 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. 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. [Ne II]12.8 μm) 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. [Ne V]14.3 μm and [O IV]25.9 μm). Based on observations collected at the European Southern Observatory, Chile, programmes 070.B-0393, 088.D-0005 and 088.B-0809.