We present Atacama Large Millimeter/submillimeter Array (ALMA) observations of the [C ii] fine-structure line and the underlying far-infrared (FIR) dust continuum emission in J1120+0641, the most distant quasar currently known ($z=7.1$). We also present observations targeting the CO(2–1), CO(7–6), and [C i] 369 μm lines in the same source obtained at the Very Large Array and Plateau de Bure Interferometer. We find a [C ii] line flux of ${F}_{[{\rm{C}}{\rm{II}}]}=1.11\pm 0.10$ Jy $\mathrm{km}\,{{\rm{s}}}^{-1}$ and a continuum flux density of ${S}_{227\mathrm{GHz}}=0.53\pm 0.04$ mJy beam−1, consistent with previous unresolved measurements. No other source is detected in continuum or [C ii] emission in the field covered by ALMA (~ 25''). At the resolution of our ALMA observations (0farcs23, or 1.2 kpc, a factor of ~70 smaller beam area compared to previous measurements), we find that the majority of the emission is very compact: a high fraction (~80%) of the total line and continuum flux is associated with a region 1–1.5 kpc in diameter. The remaining ~20% of the emission is distributed over a larger area with radius lesssim4 kpc. The [C ii] emission does not exhibit ordered motion on kiloparsec scales: applying the virial theorem yields an upper limit on the dynamical mass of the host galaxy of $(4.3\pm 0.9)\times {10}^{10}$ ${M}_{\odot }$, only ~20 × higher than the central black hole (BH). The other targeted lines (CO(2–1), CO(7–6), and [C i]) are not detected, but the limits of the line ratios with respect to the [C ii] emission imply that the heating in the quasar host is dominated by star formation, and not by the accreting BH. The star formation rate (SFR) implied by the FIR continuum is 105–340 ${M}_{\odot }\,{\mathrm{yr}}^{-1}$, with a resulting SFR surface density of ~100–350 ${M}_{\odot }\,{\mathrm{yr}}^{-1}$ kpc−2, well below the value for Eddington-accretion-limited star formation.