In the present paper, we study the terminal stage of the capillary thinning of a polymer solution jet formed with rod-like molecules. On long scales exceeding the rod length a uniform jet gets unstable according to the classical Plateau-Rayleigh pinching mechanism. We show, however, that a qualitatively different faster process, which can prevent the jet from breaking-up, generically comes into play once the jet radius becomes smaller than the rod length. Namely the solvent drains out onto the jet surface forming annular droplets there, while the rods stay trapped inside the jet polymer core. As a result, the jet core becomes more concentrated and can solidify eventually. This process can provide a universal mechanism of the capillary-induced solvent/polymer phase separation leading to fiber formation (fiber spinning) from rod-like polymer solution jets.