We have discovered that the extremely red, low-gravity L7 dwarf 2MASS J11193254−1137466 is a 0.14 ′′ (3.6 AU) binary using Keck laser guide star adaptive optics imaging. 2MASS J11193254−1137466 has previously been identified as a likely member of the TW Hydrae Association (TWA). Using our updated photometric distance and proper motion, a kinematic analysis based on the BANYAN II model gives an 82% probability of TWA membership. At TWA's 10 ± 3 Myr age and using hot-start evolutionary models, 2MASS J11193254−1137466AB is a pair of 3.7 +1.2 −0.9 M Jup brown dwarfs, making it the lowest-mass binary discovered to date. We estimate an orbital period of 90 +80 −50 years. One component is marginally brighter in K band but fainter in J band, making this a probable flux-reversal binary, the first discovered with such a young age. We also imaged the spectrally similar TWA L7 dwarf WISEA J114724.10−204021.3 with Keck and found no sign of binarity. Our evolutionary model-derived T eff estimate for WISEA J114724.10−204021.3 is ≈230 K higher than for 2MASS J11193254−1137466AB, at odds with their spectral similarity. This discrepancy suggests that WISEA J114724.10−204021.3 may actually be a tight binary with masses and temperatures very similar to 2MASS J11193254−1137466AB, or further supporting the idea that near-infrared spectra of young ultracool dwarfs are shaped by factors other than temperature and gravity. 2MASS J11193254−1137466AB will be an essential benchmark for testing evolutionary and atmospheric models in the young planetary-mass regime.