Interactions between charged macromolecules (e.g. proteins, nucleic acids, polyelectrolytes) and charged surfaces govern many natural and industrial processes. We investigate here the influence of an applied electric potential on the adsorption of charged polymers, and report the following significant result: the adsorption of certain amine side chain-containing polypeptides may become continuous, i.e. asymptotically linear (or nearly linear) in time over hours, upon the application of a modest anodic potential. Employing optical waveguide lightmode spectroscopy (OWLS) and an indium tin oxide (ITO) substrate, we show asymptotic kinetics -- and the adsorbed mass at the onset of the asymptotic regime -- to depend sensitively on polymer chemistry (in particular, side chain volume and charge location), to increase with applied potential and ionic strength (conditions favoring a thicker initial layer), and to be independent of bulk polymer concentration (suggesting post-adsorption events to be rate limiting). X-ray photoelectron spectra reveal a suppressed polymer charge within layers formed via continuous adsorption, but no evidence of electrochemical reactions. Multi-step experiments reveal post-adsorption events to lead to a gradual decrease in interfacial charge when the bulk solution is replaced with a polymer-free solution. AFM images of adsorbed layers formed under an applied potential reveal a characteristic feature size of about 30 nm. We propose a mechanism based on polymer-polymer binding within the adsorbed layer, enabled by suppressed electrostatic repulsion and/or enhanced ionic correlations near the conducting surface, and stabilized by short-range attractive interactions. Continuous adsorption under an applied electric potential offers the possibility of nanoscale films of tailored polymer content realized in a single step.