  Any particle dark matter (DM) scenario featuring a suppressed power spectrum
of astrophysical relevance results in a delay of galaxy formation. As a
consequence, such scenarios can be constrained using the global 21-cm
absorption signal initiated by the UV radiation of the first stars. The
Experiment to Detect the Global Epoch of Reionization Signature (EDGES)
recently reported the first detection of such an absorption signal at redshift
$\sim 17$. While its amplitude might indicate the need for new physics, we
solely focus on the timing of the signal to test non-cold DM models. Assuming
conservative limits for the stellar-to-baryon fraction ($f_{*}<0.03$) and for
the minimum cooling temperature ($T_{\rm vir}>10^3$ Kelvin) motivated by
radiation-hydrodynamic simulations, we are able to derive unprecedented
constraints on a variety of non-cold DM models. For example, the mass of
thermal warm DM is limited to $m_{\rm TH}>6.1$ keV, while mixed DM scenarios
(featuring a cold and a hot component) are constrained to a hot DM fraction
below 17 percent. The ultra-light axion DM model is limited to masses
$m_{a}>8\times10^{-21}$ eV, a regime where its wave-like nature is pushed far
below the kiloparsec scale. Finally, sterile neutrinos from resonant production
can be fully disfavoured as a dominant DM candidate. The results of this paper
show that the 21-cm absorption signal is a powerful discriminant of non-cold
dark matter, allowing for significant improvements over to the strongest
current limits. Confirming the result from EDGES is paramount in this context.
