  We present deep (265 ks) Chandra X-ray observations of PSO
J352.4034$-$15.3373, a quasar at z=5.831 that, with a radio-to-optical flux
ratio of R>1000, is one of the radio-loudest quasars in the early universe and
is the only quasar with observed extended radio jets of kpc-scale at $z \gtrsim
6$. Modeling the X-ray spectrum of the quasar with a power law, we find a best
fit of $\Gamma = 1.99^{+0.29}_{-0.28}$, leading to an X-ray luminosity of
$L_{2-10} = 1.26^{+0.45}_{-0.33} \times 10^{45}\ {\rm erg}\ {\rm s}^{-1}$ and
an X-ray to UV brightness ratio of $\alpha_{\rm OX} = -1.36 \pm 0.11$. We
identify a diffuse structure 50 kpc (${\sim}8^{\prime\prime}$) to the NW of the
quasar along the jet axis that corresponds to a $3\sigma$ enhancement in the
angular density of emission and can be ruled out as a background fluctuation
with a probability of P=0.9985. While with few detected photons the spectral
fit of the structure is uncertain, we find that it has a luminosity of
$L_{2-10}\sim10^{44}\ {\rm erg}\ {\rm s}^{-1}$. These observations therefore
potentially represent the most distant quasar jet yet seen in X-rays. We find
no evidence for excess X-ray emission where the previously-reported radio jets
are seen (which have an overall linear extent of $0.^{\prime\prime}28$), and a
bright X-ray point source located along the jet axis to the SE is revealed by
optical and NIR imaging to not be associated with the quasar.
