Aims. We present an analysis of the 700 ks Chandra ACIS-S observation of the field around the radio galaxy J1140-2629 (the Spiderweb Galaxy) at z = 2.156, focusing on the nuclear activity in the associated large-scale environment. Methods. We identified unresolved X-ray sources in the field down to flux limits of 1.3 x 10(-16) and 3.9 x 10(-16) erg s(-1) cm(-2) in the soft (0.5-2.0 keV) and hard (2-10 keV) band, respectively. We searched for counterparts in the optical, near-infrared, and submillimeter catalogs available in the literature to identify X-ray sources belonging to the protocluster and derived their X-ray properties. Results. 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 has a photometric redshift consistent with this range. The X-ray-emitting protocluster members are distributed approximately over a similar to 3.2 x 1.3 Mpc(2) rectangular region. An X-ray spectral analysis for all the sources within the protocluster shows that their intrinsic spectral slope is consistent with an average & lang;gamma & rang;similar to 1.84 +/- 0.04. Excluding the Spiderweb Galaxy, the best-fit intrinsic absorption for five protocluster X-ray members is N-H > 10(23) cm(-2), while another six have upper limits of the order of a few times 10(22) cm(-2). Two sources can only be fitted with very flat gamma <= 1, and are therefore considered Compton-thick candidates. The 0.5-10 keV rest-frame luminosities of the 11 Compton-thin protocluster members corrected for intrinsic absorption are greater than 2 x 10(43) erg s(-1). These values are typical for the bright end of a Seyfert-like distribution and significantly greater than X-ray luminosities expected from star formation activity. The X-ray luminosity function of the AGN in the volume associated to the Spiderweb protocluster in the range 10(43) < L-X < 10(44.5) erg s(-1) is at least ten times higher than that in the field at the same redshift and significantly flatter, implying an increasing excess at the bright end. The X-ray AGN fraction is measured to be 25.5 +/- 4.5% of the spectroscopically confirmed members in the stellar mass range log(M-*/M-?) > 10.5. This value corresponds to an enhancement factor of 6.0(-3.0)(+9.0) for the nuclear activity with L0.5 - 10 keV > 4 x 10(43) erg s(-1) with respect to the COSMOS field at comparable redshifts and stellar mass range. Conclusions. We conclude that the galaxy population in the Spiderweb protocluster is characterized by enhanced X-ray nuclear activity triggered by environmental effects on megaparsec scales.