  (Abridged) We present an analysis of the 700 ks Chandra ACIS-S observation of
the field around the Spiderweb Galaxy at z=2.156, focusing on the nuclear
activity in the associated large-scale environment. We identify unresolved
X-ray sources down to flux limits of 1.3X10^{-16} and 3.9X10^{-16} erg/s/cm^2
in the soft and hard band, respectively. We search for counterparts in the
optical, NIR and submm bands to identify X-ray sources belonging to the
protocluster. We detect 107 X-ray unresolved sources within 5 arcmin
(corresponding to 2.5 Mpc) of J1140-2629, among which 13 have optical
counterparts with spectroscopic redshift 2.11<z<2.20, and 1 source with
photometric redshift consistent with this range. Our X-ray spectral analysis
shows that their intrinsic spectral slope is consistent with an average
<\Gamma>~1.84+-0.04. The best-fit intrinsic absorption for 5 protocluster X-ray
members is N_H>10^{23} cm^{-2}, while other 6 have upper limits of the order of
fewX10^{22} cm^{-2}. Two sources can only be fitted with very flat \Gamma<=1,
and are therefore considered Compton-thick candidates. Their 0.5-10 keV rest
frame luminosities are larger than 2X10^{43} erg/s, significantly greater than
X-ray luminosities expected from star formation activity. The X-ray luminosity
function of AGN in the volume associated to the Spiderweb protocluster in the
range 10^{43}<L_X<10^{44.5} erg/s, is at least 10 times higher than that in the
field at the same redshift and significantly flatter. The X-ray AGN fraction is
measured to be (25.5+-4.5)% in the stellar mass range log(M*/M_sun)>10.5,
corresponding to an enhancement of 6.0^{+9.0}_{-3.0} with respect to the COSMOS
field at comparable redshifts and stellar mass range. We conclude that the
galaxy population in the Spiderweb Protocluster is characterized by enhanced
X-ray nuclear activity triggered by environmental effects on Mpc scales.
