The quasar ULAS J1120+0641 at redshift z = 7.085 has a highly ionized near zone which is smaller than those around quasars of similar luminosity at z ≃ 6. The spectrum also exhibits evidence for a damping wing extending redward of the systemic Lyα redshift. We use radiative transfer simulations in a cosmological context to investigate the implications for the ionization state of the inhomogeneous intergalactic medium (IGM) surrounding this quasar. Our simulations show that the transmission profile is consistent with an IGM in the vicinity of the quasar with a volume averaged H I fraction of ? and that ULAS J1120+0641 has been bright for 10^6-10^7 yr. The observed spectrum is also consistent with smaller IGM neutral fractions, ?, if a damped Lyα system in an otherwise highly ionized IGM lies within 5 proper Mpc of the quasar. This is, however, predicted to occur in only ˜5 per cent of our simulated sightlines for a bright phase of 10^6-10^7 yr. Unless ULAS J1120+0641 grows during a previous optically obscured phase, the low age inferred for the quasar adds to the theoretical challenge of forming a 2 × 10^9 Msun black hole at this high redshift.