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 $m_{chi}c^{2}lesssim 3times 10^{-2} mathrm{GeV}$, where the cosmic number density of Pop III hosts at redshift $zsim 20$ 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 $zgtrsim 20$, 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 $sigma_{1}=8times 10^{-20} mathrm{cm^{2}}$, capable of accommodating the measured absorption depth, the number density of Pop III hosts is reduced by a factor of $3-10$ at $zsim 15-20$, 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.
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