Bismuth produces different types of ordered superstructures on the InAs(100) surface, depending on the growth procedure and coverage. The (2 × 1) phase forms at completion of one Bi monolayer and consists of a uniformly oriented array of parallel lines of Bi dimers. Scanning tunneling and core level spectroscopies demonstrate its metallic character, in contrast with the semiconducting properties expected on the basis of the electron counting principle. The weak electronic coupling among neighboring lines gives rise to quasi one-dimensional Bi-derived bands with open contours at the Fermi level. Spin- and angle-resolved photoelectron spectroscopy reveals a giant Rashba splitting of these bands, in good agreement with ab initio electronic structure calculations. The very high density of the dimer lines, the metallic and quasi one-dimensional band dispersion and the Rashba-like spin texture make the Bi/InAs(100)-(2 × 1) phase an intriguing system, where novel transport regimes can be studied.