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We present an empirical model of age-dependent photospheric lithium depletion, calibrated using a large homogeneously analysed sample of 6200 stars in 52 open clusters, with ages from 2 to 6000 Myr and -0.3 < [Fe/H] < 0.2, observed in the Gaia-ESO spectroscopic survey. The model is used to obtain age estimates and posterior age probability distributions from measurements of the Li i 6708 angstrom equivalent width for individual (pre) main-sequence stars with 3000 < T-eff/K < 6500, a domain where age determination from the HR diagram is either insensitive or highly model-dependent. In the best cases, precisions of 0.1 dex in log age are achievable; even higher precision can be obtained for coeval groups and associations where the individual age probabilities of their members can be combined. The method is validated on a sample of exoplanet-hosting young stars, finding agreement with claimed young ages for some, but not others. We obtain better than 10 per cent precision in age, and excellent agreement with published ages, for seven well-studied young moving groups. The derived ages for young clusters (<1 Gyr) in our sample are also in good agreement with their training ages, and consistent with several published model-insensitive lithium depletion boundary ages. For older clusters, there remain systematic age errors that could be as large as a factor of 2. There is no evidence to link these errors to any strong systematic metallicity dependence of (pre) main-sequence lithium depletion, at least in the range -0.29 < [Fe/H] < 0.18. Our methods and model are provided as software - 'Empirical AGes from Lithium Equivalent widthS' (eagles).