  We report on the direct observation of an extended X-ray jet in the $z$=6.1
radio-loud Active Galactic Nucleus PSO J030947.49+271757.31 from a deep Chandra
X-ray observation (128 ksec). This detection represents the most distant kpc
off-nuclear emission resolved in the X-rays to date. The angular distance of
the emission is $\sim$4" (corresponding to $\sim$20 kpc at $z$=6.1), along the
same direction of the jet observed at parsec scales in previous VLBA
high-resolution radio observations. Moreover, the 0.5-7.0 keV isophotes
coincide with the extended radio emission as imaged by the VLA Sky Survey at 3
GHz. The rest-frame 2-10 keV luminosity of the extended component is
L$_{2-10keV}$=5.9$\times$10$^{44}$ erg s$^{-1}$, about 8% of the core: this
makes it one of the most luminous jets resolved in the X-rays so far. Through
Spectral Energy Distribution modelling we find that this emission can be
explained by the Inverse Compton interaction with the photons of the Cosmic
Microwave Background assuming that the jet's physical parameters are similar to
those in the local Universe. At the same time, we find that the radiation
produced by a putative population of high-energetic electrons through the
synchrotron process observed at low redshift is quenched at high redshift,
hence becoming negligible.
