Context. Recently, the high-energy (HE, 0.1–100 GeV) γ-ray emission from the object LMC P3 in the Large Magellanic Cloud (LMC) has been discovered to be modulated with a 10.3-day period, making it the first extra-galactic γ-ray binary. Aims. This work aims at the detection of very-high-energy (VHE, >100 GeV) γ-ray emission and the search for modulation of the VHE signal with the orbital period of the binary system. Methods. LMC P3 has been observed with the High Energy Stereoscopic System (H.E.S.S.); the acceptance-corrected exposure time is 100 h. The data set has been folded with the known orbital period of the system in order to test for variability of the emission. Results. VHE γ-ray emission is detected with a statistical significance ...