  We derive new constraints on the non-gravitational baryon-dark-matter
scattering (BDMS) by evaluating the mass thresholds of dark matter (DM) haloes
in which primordial gas can cool efficiently to form Population III (Pop III)
stars, based on the timing of the observed 21-cm absorption signal. We focus on
the BDMS model with interaction cross-section $\sigma=\sigma_{1}[v/(1\
\mathrm{km\ s^{-1}})]^{-4}$, where $v$ is the relative velocity of the
encounter. Our results rule out the region in parameter space with
$\sigma_{1}\gtrsim 10^{-19}\ \mathrm{cm^{2}}$ and DM particle mass
$m_{\chi}c^{2}\lesssim 3\times 10^{-2}\ \mathrm{GeV}$, where the cosmic number
density of Pop III hosts at redshift $z\sim 20$ is at least three orders of
magnitude smaller than in the standard Lambda cold DM ($\Lambda$CDM) case. In
these BDMS models, the formation of Pop III stars is significantly suppressed
for $z\gtrsim 20$, inconsistent with the timing of the observed global 21-cm
absorption signal. For the fiducial BDMS model with $m_{\chi}c^{2}=0.3$ GeV and
$\sigma_{1}=8\times 10^{-20}\ \mathrm{cm^{2}}$, capable of accommodating the
measured absorption depth, the number density of Pop III hosts is reduced by a
factor of $3-10$ at $z\sim 15-20$, when the 21-cm signal is imprinted, compared
with the $\Lambda$CDM model. The confluence of future detailed cosmological
simulations with improved 21-cm observations promises to probe the
particle-physics nature of DM at the small-scale frontier of early structure
formation.
