  Sterile neutrinos only interact with the Standard Model through the neutrino
sector, and thus represent a simple dark matter (DM) candidate with many
potential astrophysical and cosmological signatures. Recently, sterile
neutrinos produced through self-interactions of active neutrinos have received
attention as a particle candidate that can yield the entire observed DM relic
abundance without violating the most stringent constraints from X-ray
observations. We examine consistency of this production mechanism with the
abundance of small-scale structure in the universe, as captured by the
population of ultra-faint dwarf galaxies orbiting the Milky Way, and derive a
lower bound on the sterile-neutrino particle mass of $37$ keV. Combining these
results with previous limits from particle physics and astrophysics excludes
$100\%$ sterile neutrino DM produced by strong neutrino self-coupling, mediated
by a heavy ($\gtrsim 1~\mathrm{GeV}$) scalar particle; however, data permits
sterile-neutrino DM production via a light mediator.
