  Structures in Warm Dark Matter (WDM) models are exponentially suppressed
below a certain scale, characterized by the dark matter particle mass, $m_{\rm
x}$. Since structures form hierarchically, the presence of collapsed objects at
high-redshifts can set strong lower limits on $m_{\rm x}$. We place robust
constraints on $m_{\rm x}$ using recent results from the {\it Swift} database
of high-redshift gamma-ray bursts (GRBs). We parameterize the redshift
evolution of the ratio between the cosmic GRB rate and star formation rate
(SFR) as $\propto (1+z)^\alpha$, thereby allowing astrophysical uncertainties
to partially mimic the cosmological suppression of structures in WDM models.
Using a maximum likelihood estimator on two different $z>4$ GRB subsamples
(including two bursts at $z>8$), we constrain $m_{\rm x} \gtrsim 1.6$-1.8 keV
at 95% C.L., when marginalized over a flat prior in $\alpha$. We further
estimate that 5 years of a SVOM-like mission would tighten these constraints to
$m_{\rm x} \gtrsim 2.3 $ keV. Our results show that GRBs are a powerful probe
of high-redshift structures, providing robust and competitive constraints on
$m_{\rm x}$.
