  PSR J1357-6429 is a Vela-like radio pulsar that has been recently detected
with Chandra and Fermi, which, like Vela, powers a compact X-ray pulsar wind
nebula and X-ray-radio plerion associated with an extended TeV source. We
present our deep optical observations with the Very Large Telescope to search
for an optical counterpart of the pulsar and its nebula. We detected a
point-like source in V, R, and I bands whose position is in agreement with the
X-ray position of the pulsar, and whose colours are distinct from those of
ordinary stars. The tentative optical luminosity and efficiency of the source
are similar to those of the Vela pulsar, which also supports the optical
identification. However, the source spectrum is unusually steep, with a
spectral index of about 5, which is not typical of optical pulsars. The source
offset from the radio position of PSR J1357-6429, which is in line with the
corresponding offset of the X-ray position, implies the pulsar transverse
velocity of 1600-2000 km/s at the distance of 2-2.5 kpc, making it the fastest
moving pulsar known.
