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 kiloparsec scale at z ≳ 6. Modeling the X-ray spectrum of the quasar with a power law, we find a best fit of
Γ
=
1.99
−
0.28
+
0.29
, leading to an X-ray luminosity of
L
2
–
10
=
1.26
−
0.33
+
0.45
×
10
45
erg
s
−
1
and an X-ray to UV brightness ratio of α
OX = −1.45 ± −0.11. We identify a diffuse structure 50 kpc (∼8″) to the NW of the quasar along the jet axis that corresponds to a 3σ 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 ∼ 1044 erg 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.″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.