We present the results of an ultradeep, narrowband imaging survey for Lyman-continuum (LyC) emission at z~3 in the SSA22a field. We employ a custom narrowband filter centered at 3640A (NB3640), which probes the LyC region for galaxies at z>3.06. We also analyze new and archival NB4980 imaging tuned to the wavelength of the Lya emission line at z=3.09, and archival broadband B, V, and R images of the non-ionizing UV continuum. Our NB3640 images contain 26 z>3.06 Lyman Break Galaxies (LBGs) as well as a set of 130 Lya emitters (LAEs), identified by their excess NB4980 flux relative to the BV continuum. Six LBGs and 28 LAEs are detected in the NB3640 image. LBGs appear to span a range of NB3640-R colors, while LAEs appear bimodal in their NB3640-R properties. We estimate average UV to LyC flux density ratios, corrected for foreground contamination and intergalactic medium absorption, finding <F_{UV}/F_{LyC}>^{LBG} = 11.3^{+10.3}_{-5.4}, which implies a LBG LyC escape fraction f_{esc}^{LyC} ~ 0.1, and <F_{UV}/F_{LyC}>^{LAE} = 2.2^{+0.9}_{-0.6}. The strikingly blue LAE flux density ratios defy interpretation in terms of standard stellar population models. Assuming <F_{UV}/F_{LyC}>^{LBG} applies down to L=0.1L*, we estimate a galaxy contribution to the intergalactic hydrogen ionization rate that is consistent with independent estimates based on the Lya forest opacity at z~3. If we assume that <F_{UV}/F_{LyC}>^{LAE} holds at the faintest luminosities, the galaxy contribution significantly exceeds that inferred from the Lya forest. Further follow-up study of these faint LAEs is crucial, given the potentially important contribution similar objects make to the process of reionization. (Abridged)
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