The evolution of sequence-defined synthetic polymers made of building blocks beyond those compatible with polymerase enzymes or the ribosome has the potential to generate new classes of receptors, catalysts and materials. Here we describe a ligase-mediated DNA-templated polymerization and in vitro selection system to evolve highly functionalized nucleic acid polymers (HFNAPs) made from 32 building blocks that contain eight chemically diverse side chains on a DNA backbone. Through iterated cycles of polymer translation, selection and reverse translation, we discovered HFNAPs that bind proprotein convertase subtilisin/kexin type 9 (PCSK9) and interleukin-6, two protein targets implicated in human diseases. Mutation and reselection of an active PCSK9-binding polymer yielded evolved polymers with high affinity (KD = 3 nM). This evolved polymer potently inhibited the binding between PCSK9 and the low-density lipoprotein receptor. Structure-activity relationship studies revealed that specific side chains at defined positions in the polymers are required for binding to their respective targets. Our findings expand the chemical space of evolvable polymers to include densely functionalized nucleic acids with diverse, researcher-defined chemical repertoires.