  We present high resolution X-ray observations of the narrow line radio galaxy
PKS 1138-262 at z = 2.156 with the ACIS-S detector on the Chandra observatory.
These observations show that the X-ray emission from 1138-262 is dominated by
emission from the AGN with a (rest frame) 2 to 10 keV luminosity of 4x10^{45}
erg/s. The relative X-ray and radio properties of the AGN in 1138-262 are
similar to those seen for the AGN in the archetype powerful radio galaxy Cygnus
A. Between 10% and 25% (depending on energy) of the X-ray emission from
1138--262 is spatially extended on scales of 10'' to 20''. The extended X-ray
emission is elongated, with a major axis aligned with that of the radio source.
While the X-ray and radio emissions are elongated on similar scales and
position angles, there is no one-to-one correspondence between the radio and
X-ray features in the source. The most likely origin for the extended X-ray
emission in 1138-262 is thermal emission from shocked gas, although we cannot
rule-out a contribution from inverse Compton emission. If the emission is from
hot gas, the gas density is 0.05 cm^{-3} and the gas mass is 2.5x10^{12}
M_solar. The pressure in this hot gas is adequate to confine the radio emitting
plasma and the optical line emitting gas. We set an upper limit of 1.5x10^{44}
erg/s to the 2 to 10 keV luminosity of any normal cluster atmosphere associated
with 1138-262. No emission was detected from any of the Ly alpha emitting
galaxies in the (proto-) cluster around 1138-262, outside of the Ly alpha halo
of 1138-262 itself, to a 2 to 10keV luminosity limit of 1.2x10^{43}erg/s.
