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[Abridged] We investigate the physical properties of a Lyman continuumemitter candidate at $z=3.212$ with photometric coverage from $U$ to MIPS24$\mu$m band and VIMOS/VLT and MOSFIRE/Keck spectroscopy. Investigation of theUV spectrum confirms a direct spectroscopic detection of the Lyman continuumemission with $S/N>5$. Non-zero Ly$\alpha$ flux at the systemic redshift andhigh Lyman-$\alpha$ escape fraction suggest a low HI column density. The weak Cand Si low-ionization absorption lines are also consistent with a low coveringfraction along the line of sight. The[OIII]$\lambda\lambda4959,5007+\mathrm{H}\beta$ equivalent width is one of thelargest reported for a galaxy at $z>3$($\mathrm{EW}([\mathrm{OIII}]\lambda\lambda4959,5007+\mathrm{H}\beta) \simeq1600\AA$, rest-frame) and the NIR spectrum shows that this is mainly due to anextremely strong [OIII] emission. The large observed [OIII]/[OII] ratio ($>10$)and high ionization parameter are consistent with prediction fromphotoionization models in case of a density-bounded nebula scenario.Furthermore, the$\mathrm{EW}([\mathrm{OIII}]\lambda\lambda4959,5007+\mathrm{H}\beta)$ iscomparable to recent measurements reported at $z\sim7-9$, in the reionizationepoch. We also investigate the possibility of an AGN contribution to explainthe ionizing emission but most of the AGN identification diagnostics suggestthat stellar emission dominates instead. This source is currently the firsthigh-$z$ example of a Lyman continuum emitter exhibiting indirect and directevidences of a Lyman continuum leakage and having physical propertiesconsistent with theoretical expectation from Lyman continuum emission from adensity-bounded nebula.

An extreme [OIII] emitter at $z=3.2$: a low metallicity Lyman continuum source