  Vela X is a region of extended radio emission in the western part of the Vela
constellation: one of the nearest pulsar wind nebulae (PWNe), and associated
with the energetic Vela pulsar (PSR B0833-45). Extended very-high-energy (VHE)
$\gamma$-ray emission (HESS $\mathrm{J0835\mhyphen 455}$) was discovered using
the H.E.S.S. experiment in 2004. The VHE $\gamma$-ray emission was found to be
coincident with a region of X-ray emission discovered with ${\it ROSAT}$ above
1.5 keV (the so-called \textit{Vela X cocoon}): a filamentary structure
extending southwest from the pulsar to the centre of Vela X. A deeper
observation of the entire Vela X nebula region, also including larger offsets
from the cocoon, has been performed with H.E.S.S. This re-observation was
carried out in order to probe the extent of the non-thermal emission from the
Vela X region at TeV energies and to investigate its spectral properties. In
order to increase the sensitivity to the faint $\gamma$-ray emission from the
very extended Vela X region, a multivariate analysis method combining three
complementary reconstruction techniques of Cherenkov-shower images is applied
for the selection of $\gamma$-ray events. The analysis is performed with the
On/Off background method, which estimates the background from separate
observations pointing away from Vela X; towards regions free of $\gamma$-ray
sources but with comparable observation conditions. The $\gamma$-ray surface
brightness over the large Vela X region reveals that the detection of
non-thermal VHE $\gamma$-ray emission from the PWN HESS $\mathrm{J0835\mhyphen
455}$ is statistically significant over a region of radius 1.2$^{\circ}$ around
the position $\alpha$ = 08$^{\mathrm{h}}$ 35$^{\mathrm{m}}$ 00$^{\mathrm{s}}$,
$\delta$ = -45$^{\circ}$ 36$^{\mathrm{\prime}}$
00$^{\mathrm{\prime}\mathrm{\prime}}$ (J2000).
