Embedded AGN and star formation in the central 80 pc of IC 3639

[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 $90 \pm 20\, \rm{mJy}$ at $10.5\, \rm{\mu m}$, measured with three different baselines in VLTI (UT1-UT2, UT3-UT4, and UT2-UT3; $46$-$58\, \rm{m}$), 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 $80\, \rm{pc}$ ($40$-$340\, \rm{mas}$). 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]$_{12.8 \rm{\mu m}}$) with a net contribution of $\sim 70\%$. The AGN accounts for the remaining $\sim 30\%$ of the mid-IR flux, ascribed to the unresolved component in the MIDI observations, and the ionisation of high-excitation lines (e.g. [NeV]$_{14.3 \rm{\mu m}}$ and [OIV]$_{25.9 \rm{\mu m}}$).