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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., Comment: Accepted for publication on A&A