The electronic subband structure of carriers in bismuth-like cylindrical nanowires is investigated analytically, using an anisotropic effective mass model. Quantum confinement effects are found to be significantly enhanced over the results of a quasi-isotropic approach, due to the correct consideration of both the mass anisotropy and the boundary shape of the wire. The size quantization problem for a carrier with anisotropic effective mass parameters in a cylindrical well is shown to be equivalent to that of a carrier with some isotropic effective mass in an elliptical well. Detailed study of the energy levels reveals their orbital degeneracy is lifted by the elliptic symmetry, where the degree of ellipticity corresponds to the mass anisotropy. Carrier motion is analysed by analogy to the geometrical optics of elliptic waveguides, with bounding caustic curves defining two groups inside the wire that correspond to 'whispering gallery' and 'jumping ball' modes. The additional confinement, arising from the mass anisotropy, leads to larger critical wire diameters for the semimetal-to-semiconductor transition, which is investigated for wires of different crystallographic orientation.