We present the first simultaneous, multiwavelength observations of an L dwarf, the L3.5 candidate brown dwarf 2MASS J00361617+1821104, conducted with the Very Large Array, the Chandra X-Ray Observatory, and the Kitt Peak 4 m telescope. We detect strongly variable and periodic radio emission (P=3 hr) with a fraction of about 60% circular polarization. No X-ray emission is detected to a limit of LX/Lbol<~2×10-5, several hundred times below the saturation level observed in early M dwarfs. Similarly, we do not detect Hα emission to a limit of LHα/Lbol<~2×10-7, the deepest for any L dwarf observed to date. The ratio of radio to X-ray luminosity is at least 4 orders of magnitude in excess of that observed in a wide range of active stars (including M dwarfs), providing the first direct confirmation that late-M and L dwarfs violate the radio/X-ray correlation. The radio emission is due to gyrosynchrotron radiation in a large-scale magnetic field of about 175 G, which is maintained on timescales longer than 3 yr. The detected 3 hr period may be due to (1) the orbital motion of a companion at a separation of about 5 stellar radii, similar to the configuration of RS CVn systems, (2) an equatorial rotation velocity of about 37 km s-1 and an anchored, long-lived magnetic field, or (3) periodic release of magnetic stresses in the form of weak flares. In the case of orbital motion, the magnetic activity may be induced by the companion, possibly explaining the unusual pattern of activity and the long-lived signal. We conclude that fully convective stars can maintain a large-scale and stable magnetic field, but the lack of X-ray and Hα emission indicates that the atmospheric conditions are markedly different than in early-type stars and even M dwarfs. Similar observations are therefore invaluable for probing both the internal and external structure of low-mass stars and substellar objects, and for providing constraints on dynamo models.