The recent discovery of possible high-temperature (T-c) superconductivity over 65 K in a monolayer FeSe film on SrTiO3 (refs 1-6) triggered a fierce debate on how superconductivity evolves from bulk to film, because bulk FeSe crystal exhibits a T-c of no higher than 10 K (ref. 7). However, the difficulty in controlling the carrier density and the number of FeSe layers has hindered elucidation of this problem(4,8). Here, we demonstrate that deposition of potassium onto FeSe films markedly expands the accessible doping range towards the heavily electron-doped region. Intriguingly, we have succeeded in converting non-superconducting films with various thicknesses into superconductors with T-c as high as 48 K. We also found a marked increase in the magnitude of the superconducting gap on decreasing the FeSe film thickness, indicating that the interface plays a crucial role in realizing the high-temperature superconductivity. The results presented provide a new strategy to enhance and optimize T-c in ultrathin films of iron-based superconductors.