We present the first measurements of the Lyman-continuum photon production efficiency $xi_{textrm{ion,0}}$ at $zsim4$-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 403r{A} to 2818r{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.
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