Context : 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 .

Aims : We present our deep optical observations with the Very Large Telescope to search for an optical counterpart of the pulsar and its nebula .

Methods : The observations were carried out using a direct imaging mode in the V , R , and I bands .
We also analysed archival X-ray data obtained with Chandra and XMM-Newton .

Results : In all three optical bands , we detect a point-like source with V = 27.3 \pm 0.3 , R = 25.52 \pm 0.07 , and I = 24.13 \pm 0.05 , 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 \alpha _ { \nu } \sim 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 .

Conclusions : The position and peculiar colours of the detected source suggest that it is an optical counterpart of the pulsar .
Further high spatial-resolution infrared observations can help to verify its real nature .