Many bacteria live in polymeric fluids, such as mucus, environmental polysaccharides, and extracellular polymers in biofilms. However, laboratory studies typically focus on cells in polymer-free fluids. Here, we show that interactions with polymers shape a fundamental feature of bacterial life-how they proliferate in space in multicellular colonies. Using experiments, we find that when polymer is sufficiently concentrated, cells generically and reversibly form large serpentine "cables" as they proliferate. By combining experiments with biophysical theory and simulations, we demonstrate that this distinctive form of colony morphogenesis arises from an interplay between polymer-induced entropic attraction between neighboring cells and their hindered ability to diffusely separate from each other in a viscous polymer solution. Our work thus reveals a pivotal role of polymers in sculpting proliferating bacterial colonies, with implications for how they interact with hosts and with the natural environment, and uncovers quantitative principles governing colony morphogenesis in such complex environments.