Scattering between dark matter (DM) and protons leads to suppressed small-scale fluctuations, with implications for a variety of cosmological observables. In this Letter, we search for evidence of DM-proton scattering with an interaction cross section $\sigma ={\sigma }_0(v/c{)}^n$ for $n=0$, 2 and 4, corresponding, e.g., to velocity-independent contact interactions from heavy mediators, velocity-dependent pseudoscalar-mediated scattering, and higher-order dipole interactions, respectively, using high-redshift ($z\sim 4–10$) ultraviolet galaxy luminosity functions (UVLFs) observed by the Hubble Space Telescope (HST). We employ an adjusted implementation of gallumi combined with the modified Boltzmann solver class dmeff that accounts for interacting DM, and incorporate UVLF data from both blank and lensed HST fields, alongside Planck CMB data and the Pantheon supernova catalog in a Bayesian analysis framework to set constraints on ${\sigma }_0$. Our results show that including lensed UVLF data, which probe fainter galaxies than the blank HST fields and thus smaller scales, leads to a substantial improvement in the constraints on ${\sigma }_0$ for $n>0$, surpassing existing bounds from Milky Way (MW) satellite abundance and CMB anisotropies. For $m_{\chi }=1\mathrm{MeV}$, for example, we set the upper bounds at $1.1\times {10}^{-25}{\mathrm{cm}}^2$ for $n=2$ and $2.1\times {10}^{-22}{\mathrm{cm}}^2$ for $n=4$. For $n=0$, our bound is within an order of magnitude of those from the Lyman-$\alpha$ forest and MW satellites.

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• Cosmology
• Large scale structure of the Universe
• Particle dark matter