We present a detailed analysis of the far-infrared (-IR) properties of the bright, lensed, z = 2.3, submillimetre-selected galaxy (SMG), SMM J2135-0102 (hereafter SMM J2135), using new observations with Herschel, SCUBA-2 and the Very Large Array (VLA). These data allow us to constrain the galaxy's spectral energy distribution (SED) and show that it has an intrinsic rest-frame 8-1000-μm luminosity, L_(bol), of (2.3±0.2) × 10^(12) L_☉ and a likely star-formation rate (SFR) of ~400 yr-1. The galaxy sits on the far-IR/radio correlation for far-IR-selected galaxies. At ≳70 μm, the SED can be described adequately by dust components with dust temperatures, T_d ~ 30 and 60 k. Using SPIRE's Fourier- transform spectrometer (FTS) we report a detection of the [C ii] 158 μm cooling line. If the [C ii], CO and far-IR continuum arise in photo-dissociation regions (PDRs), we derive a characteristic gas density, n ~ 10^3 cm^(-3), and a far-ultraviolet (-UV) radiation field, G_0, 10^(3)× stronger than the Milky Way. L_[CII]/L_(bol) is significantly higher than in local ultra-luminous IR galaxies (ULIRGs) but similar to the values found in local star-forming galaxies and starburst nuclei. This is consistent with SMM J2135 being powered by starburst clumps distributed across ~2 kpc, evidence that SMGs are not simply scaled-up ULIRGs. Our results show that SPIRE's FTS has the ability to measure the redshifts of distant, obscured galaxies via the blind detection of atomic cooling lines, but it will not be competitive with ground-based CO-line searches. It will, however, allow detailed study of the integrated properties of high-redshift galaxies, as well as the chemistry of their interstellar medium (ISM), once more suitably bright candidates have been found
