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We isolate a sample of 35 z \u2248 4\u20135 continuum-faint Lyman-\u03b1 emitters from deep VLT\/MUSE spectroscopy and directly measure their H\u2009\u03b1 emission using stacked Spitzer\/IRAC Ch. 1 photometry. Based on Hubble Space Telescope imaging, we determine that the average UV continuum magnitude is fainter than \u221216 (\u2248 0.01 L\u22c6), implying a median Lyman-\u03b1 equivalent width of 259 \u00c5. By combining the H\u2009\u03b1 measurement with the UV magnitude, we determine the ionizing photon production efficiency, \u03beion, a first for such faint galaxies. The measurement of log10 (\u03beion [Hz\u2009erg\u22121]) =\u00a026.28 ($^{+0.28}_{-0.40}$) is in excess of literature measurements of both continuum- and emission line-selected samples, implying a more efficient production of ionizing photons in these lower luminosity, Lyman-\u03b1-selected systems. 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