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Possible optical counterpart of PSR J1357-6429

(Abridged) PSR J1357-6429 is a Vela-like radio pulsar that has been recently detected in X-rays and gamma-rays. It powers a compact tail-like X-ray pulsar wind nebula and X-ray-radio plerion associated with an extended TeV source HESS J1356-645. We present our deep optical observations with the Very Large Telescope to search for an optical counterpart of the pulsar and its nebula. We detect a point-like source in V, R, and I bands whose position is within the 1-sigma error circle of the X-ray position of the pulsar, and whose colours are distinct from those of ordinary stars. We consider it as a candidate optical counterpart of the pulsar. If it is indeed the counterpart, its 5-sigma offset from the radio pulsar position, measured about 9 yr earlier, implies that the transverse velocity of the pulsar is in the range of 1600--2000 km s^{-1} at the distance of 2--2.5 kpc, making it the fastest moving pulsar known. The direction of the estimated proper motion coincides with the extension of the pulsar's X-ray tail, suggesting that this is a jet. The tentative optical luminosity and efficiency of the pulsar are similar to those of the Vela pulsar, which also supports the optical identification. However, the candidate undergoes an unusually steep dereddened flux increase towards the infrared with a spectral index of about 5, that is not typical of optical pulsars. It implies a strong double-knee spectral break in the pulsar emission between the optical and X-rays. The reasons for the spectral steepness are unclear. It may be caused by a nebula knot projected onto the jet and strongly overlapping with the pulsar, as observed for the Crab, where the knot has a significantly steeper spectrum than the pulsar. We find no other signs of the pulsar nebula in the optical. Alternatively, the detected source may be a faint AGN, that has not yet been seen at other wavelengths.