  We present the first measurements of the Lyman-continuum photon production
efficiency $\xi_{\textrm{ion,0}}$ at $z\sim4$-5 for galaxies fainter than 0.2
$L^*$ ($-$19 mag). $\xi_{\textrm{ion,0}}$ quantifies the production rate of
ionizing photons with respect to the UV luminosity density assuming a fiducial
escape fraction of zero. Extending previous measurements of
$\xi_{\textrm{ion,0}}$ to the faint population is important, as ultra-faint
galaxies are expected to contribute the bulk of the ionizing emissivity. We
probe $\xi_{\textrm{ion,0}}$ to such faint magnitudes by taking advantage of
200-hour depth Spitzer/IRAC observations from the GREATS program and
$\approx$300 3<$z$<6 galaxies with spectroscopic redshifts from the MUSE GTO
Deep + Wide programs. Stacked IRAC [3.6]$-$[4.5] colors are derived and used to
infer the H$\alpha$ rest-frame equivalent widths, which range from 403\r{A} to
2818\r{A}. The derived $\xi_{\textrm{ion,0}}$ is
$\log_{10}(\xi_{\textrm{ion,0}} / \textrm{Hz erg}^{-1}) = 25.36 \pm 0.08$ over
$-$20.5 < M$_{\textrm{UV}}$ < $-$17.5, similar to those derived for brighter
galaxy samples at the same redshift and therefore suggesting that
$\xi_{\textrm{ion}}$ shows no strong dependence on $M_{UV}$. The
$\xi_{\textrm{ion,0}}$ values found in our sample imply that the
Lyman-continuum escape fraction for $M_{\textrm{UV}} \approx -19$ star-forming
galaxies cannot exceed $\approx$8-20\% in the reionization era.
