We present Chandra ACIS-S and Australia Telescope Compact Array (ATCA) radio continuum observations of the strongly lensed dusty, star-forming galaxy SPT-S J034640-5204.9 (hereafter SPT0346-52) at z = 5.656. This galaxy has also been observed with ALMA, HST, Spitzer, Herschel, Atacama Pathfinder EXperiment, and the Very Large Telescope. Previous observations indicate that if the infrared (IR) emission is driven by star formation, then the inferred lensing-corrected star formation rate (SFR) (similar to 4500 M-circle dot yr(-1)) and SFR surface density Sigma(SFR) (similar to 2000 M-circle dot yr(-1) kpc(-2)) are both exceptionally high. It remained unclear from the previous data, however, whether a central active galactic nucleus (AGN) contributes appreciably to the IR luminosity. The Chandra upper limit shows that SPT0346-52 is consistent with being star formation dominated in the X-ray, and any AGN contribution to the IR emission is negligible. The ATCA radio continuum upper limits are also consistent with the FIR-to-radio correlation for star-forming galaxies with no indication of an additional AGN contribution. The observed prodigious intrinsic IR luminosity of (3.6 +/- 0.3) x 10(13) L-circle dot originates almost solely from vigorous star formation activity. With an intrinsic source size of 0.61 +/- 0.03 kpc, SPT0346-52 is confirmed to have one of the highest Sigma(SFR) of any known galaxy. This high Sigma(SFR), which approaches the Eddington limit for a radiation pressure supported starburst, may be explained by a combination of very high star formation efficiency and gas fraction.