Cross sections for elastic and inelastic scattering of the weakly bound ⁹Be nucleus on a ¹²⁰Sn target have been measured at seven bombarding energies around and above the Coulomb barrier. The elastic angular distributions are analyzed with a four-body continuum-discretized coupled-channels (CDCC) calculation, which considers ⁹Be as a three-body projectile (α + α + n). An optical model analysis using the São Paulo potential is also shown for comparison. The CDCC analysis shows that the coupling to the continuum part of the spectrum is important for the agreement with experimental data even at energies around the Coulomb barrier, suggesting that breakup is an important process at low energies. At the highest incident energies, two inelastic peaks are observed at 1.19(5) and 2.41(5) MeV. Coupled-channels (CC) calculations using a rotational model confirm that the first inelastic peak corresponds to the excitation of the 2⁺₁ state in ¹²⁰Sn, while the second one likely corresponds to the excitation of the 3^-₁ state.