  We present a search for an unstable sterile neutrino by looking for a
matter-induced signal in eight years of atmospheric $\nu_\mu$ data collected
from 2011 to 2019 at the IceCube Neutrino Observatory. Both the (stable)
three-neutrino and the 3+1 sterile neutrino models are disfavored relative to
the unstable sterile neutrino model, though with $p$-values of 2.5\% and
0.81\%, respectively, we do not observe evidence for 3+1 neutrinos with
neutrino decay. The best-fit parameters for the sterile neutrino with decay
model from this study are $\Delta m_{41}^2=6.7^{+3.9}_{-2.5}\,\rm{eV}^2$,
$\sin^2 2\theta_{24}=0.33^{+0.20}_{-0.17}$, and $g^2=2.5\pi\pm1.5\pi$, where
$g$ is the decay-mediating coupling. The preferred regions from short-baseline
oscillation searches are excluded at 90\% C.L.
