We present the identification of the COCONUTS-2 system, composed of the M3 dwarf L 34-26 and the T9 dwarf WISEPA J075108.79$-$763449.6. Given their common proper motions and parallaxes, these two field objects constitute a physically bound pair with a projected separation of 594$$ (6471 au). The primary star COCONUTS-2A has strong stellar activity (H$alpha$, X-ray, and UV emission) and is rapidly rotating ($P_{rm rot} = 2.83$ days), from which we estimate an age of 150-800 Myr. Comparing equatorial rotational velocity derived from the TESS light curve to spectroscopic $vsin{i}$, we find COCONUTS-2A has a nearly edge-on inclination. The wide exoplanet COCONUTS-2b has an effective temperature of $T_{rm eff}=434 pm 9$ K, a surface gravity of $log{g} = 4.11^{+0.11}_{-0.18}$ dex, and a mass of $M=6.3^{+1.5}_{-1.9}$ $M_{rm Jup}$ based on hot-start evolutionary models, leading to a $0.016^{+0.004}_{-0.005}$ mass ratio for the COCONUTS-2 system. COCONUTS-2b is the second coldest (after WD 0806$-$661B) and the second widest (after TYC 9486-927-1 b) exoplanet imaged to date. Comparison of COCONUTS-2bs infrared photometry with ultracool model atmospheres suggests the presence of both condensate clouds and non-equilibrium chemistry in its photosphere. Similar to 51 Eri b, COCONUTS-2b has a sufficiently low luminosity ($log{(L_{rm bol}/L_{odot})} = -6.384 pm 0.028$ dex) to be consistent with the cold-start process that may form gas-giant (exo)planets, though its large separation means such formation would not have occurred in situ. Finally, at a distance of 10.9 pc, COCONUTS-2b is the nearest imaged exoplanet to Earth known to date.
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