We present constraints on the masses of extremely light bosons dubbed fuzzy dark matter (FDM) from Lyman-α forest data. Extremely light bosons with a de Broglie wavelength of ∼1  kpc have been suggested as dark matter candidates that may resolve some of the current small scale problems of the cold dark matter model. For the first time, we use hydrodynamical simulations to model the Lyman-α flux power spectrum in these models and compare it to the observed flux power spectrum from two different data sets: the XQ-100 and HIRES/MIKE quasar spectra samples. After marginalization over nuisance and physical parameters and with conservative assumptions for the thermal history of the intergalactic medium (IGM) that allow for jumps in the temperature of up to 5000 K, XQ-100 provides a lower limit of 7.1×10^{-22}  eV, HIRES/MIKE returns a stronger limit of 14.3×10^{-22}  eV, while the combination of both data sets results in a limit of 20×10^{-22}  eV (2σ C.L.). The limits for the analysis of the combined data sets increases to 37.5×10^{-22}  eV (2σ C.L.) when a smoother thermal history is assumed where the temperature of the IGM evolves as a power law in redshift. Light boson masses in the range 1-10×10^{-22}  eV are ruled out at high significance by our analysis, casting strong doubts that FDM helps solve the "small scale crisis" of the cold dark matter models.