  Aims. Colliding wind binary systems have long been suspected to be
high-energy (HE; 100 MeV < E < 100 GeV) {\gamma}-ray emitters. {\eta} Car is
the most prominent member of this object class and is confirmed to emit
phase-locked HE {\gamma} rays from hundreds of MeV to ~100 GeV energies. This
work aims to search for and characterise the very-high-energy (VHE; E >100 GeV)
{\gamma}-ray emission from {\eta} Car around the last periastron passage in
2014 with the ground-based High Energy Stereoscopic System (H.E.S.S.). Methods.
The region around {\eta} Car was observed with H.E.S.S. between orbital phase p
= 0.78 - 1.10, with a closer sampling at p {\approx} 0.95 and p {\approx} 1.10
(assuming a period of 2023 days). Optimised hardware settings as well as
adjustments to the data reduction, reconstruction, and signal selection were
needed to suppress and take into account the strong, extended, and
inhomogeneous night sky background (NSB) in the {\eta} Car field of view.
Tailored run-wise Monte-Carlo simulations (RWS) were required to accurately
treat the additional noise from NSB photons in the instrument response
functions. Results. H.E.S.S. detected VHE {\gamma}-ray emission from the
direction of {\eta} Car shortly before and after the minimum in the X-ray
light-curve close to periastron. Using the point spread function provided by
RWS, the reconstructed signal is point-like and the spectrum is best described
by a power law. The overall flux and spectral index in VHE {\gamma} rays agree
within statistical and systematic errors before and after periastron. The
{\gamma}-ray spectrum extends up to at least ~400 GeV. This implies a maximum
magnetic field in a leptonic scenario in the emission region of 0.5 Gauss. No
indication for phase-locked flux variations is detected in the H.E.S.S. data.
