  Context. Recently, the high-energy (HE, 0.1-100 GeV) $\gamma$-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
$\gamma$-ray binary.
  Aims. This work aims at the detection of very-high-energy (VHE, >100 GeV)
$\gamma$-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. Energy spectra are obtained for the orbit-averaged
data set, and for the orbital phase bin around the VHE maximum.
  Results. VHE $\gamma$-ray emission is detected with a statistical
significance of 6.4 $\sigma$. The data clearly show variability which is
phase-locked to the orbital period of the system. Periodicity cannot be deduced
from the H.E.S.S. data set alone. The orbit-averaged luminosity in the $1-10$
TeV energy range is $(1.4 \pm 0.2) \times 10^{35}$ erg/s. A luminosity of $(5
\pm 1) \times 10^{35}$ erg/s is reached during 20% of the orbit. HE and VHE
$\gamma$-ray emissions are anti-correlated. LMC P3 is the most luminous
$\gamma$-ray binary known so far.
