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We present Chandra ACIS-S and ATCA radio continuum observations of thestrongly lensed dusty, star-forming galaxy SPT-S J034640-5204.9 (hereafterSPT0346-52) at $z$ = 5.656. This galaxy has also been observed with ALMA, HST,Spitzer, Herschel, APEX, and the VLT. Previous observations indicate that ifthe infrared (IR) emission is driven by star formation, then the inferredlensing-corrected star formation rate ($\sim$ 4500 $M_{\sun}$ yr$^{-1}$) andstar formation rate surface density $\Sigma_{\rm SFR}$ ($\sim$ 2000 $M_{\sun}{yr^{-1}} {kpc^{-2}}$) are both exceptionally high. It remained unclear fromthe previous data, however, whether a central active galactic nucleus (AGN)contributes appreciably to the IR luminosity. The {\it Chandra} upper limitshows that SPT0346-52 is consistent with being star-formation dominated in theX-ray, and any AGN contribution to the IR emission is negligible. The ATCAradio continuum upper limits are also consistent with the FIR-to-radiocorrelation for star-forming galaxies with no indication of an additional AGNcontribution. The observed prodigious intrinsic IR luminosity of (3.6 $\pm$0.3) $\times$ 10$^{13}$ $L_{\sun}$ originates almost solely from vigorous starformation activity. With an intrinsic source size of 0.61 $\pm$ 0.03 kpc,SPT0346-52 is confirmed to have one of the highest $\Sigma_{SFR}$ of any knowngalaxy. This high $\Sigma_{SFR}$, which approaches the Eddington limit for aradiation pressure supported starburst, may be explained by a combination ofvery high star formation efficiency and gas fraction.

SPT0346-52: Negligible AGN Activity in a Compact, Hyper-starburst Galaxy at z = 5.7