  If a significant fraction of the dark matter in the Universe is made of an
ultra-light scalar field, named fuzzy dark matter (FDM) with a mass $m_a$ of
the order of $10^{-22}-10^{-21}$ eV, then its de Broglie wavelength is large
enough to impact the physics of large scale structure formation. In particular,
the associated cut-off in the linear matter power spectrum modifies the
structure of the intergalactic medium (IGM) at the scales probed by the
Lyman-$\alpha$ forest of distant quasars. We study this effect by making use of
dedicated cosmological simulations which take into account the hydrodynamics of
the IGM. We explore heuristically the amplitude of quantum pressure for the FDM
masses considered here and conclude that quantum effects should not modify
significantly the non-linear evolution of matter density at the scales relevant
to the measured Lyman-$\alpha$ flux power, and for $m_a \geq 10^{-22}$ eV. We
derive a scaling law between $m_a$ and the mass of the well-studied thermal
warm dark matter (WDM) model that is best adapted to the Lyman-$\alpha$ forest
data, and differs significantly from the one infered by a simple linear
extrapolation. By comparing FDM simulations with the Lyman-$\alpha$ flux power
spectra determined from the BOSS survey, and marginalizing over relevant
nuisance parameters, we exclude FDM masses in the range $10^{-22} \leq m_a <
2.3\times 10^{-21}$ eV at 95 % CL. Adding higher-resolution Lyman-$\alpha$
spectra extends the exclusion range up to $2.9\times 10^{-21}$ eV. This
provides a significant constraint on FDM models tailored to solve the
"small-scale problems" of $\Lambda$CDM.
