  We present constraints on the masses of extremely light bosons dubbed fuzzy
dark matter from Lyman-$\alpha$ forest data. Extremely light bosons with a De
Broglie wavelength of $\sim 1$ 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-$\alpha$ flux power spectrum in these models and compare with
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 IGM that allow for jumps in the temperature of up to $5000\rm\,K$,
XQ-100 provides a lower limit of 7.1$\times 10^{-22}$ eV, HIRES/MIKE returns a
stronger limit of 14.3$\times 10^{-22}$ eV, while the combination of both data
sets results in a limit of 20 $\times 10^{-22}$ eV (2$\sigma$ C.L.). The limits
for the analysis of the combined data sets increases to 37.5$\times 10^{-22}$
eV (2$\sigma$ 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 $1-10 \times10^{-22}$ 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.
