Electrochemical energy storage is one of the main societal challenges of this century. The performances of classical lithium-ion technology based on liquid electrolytes have made great advances in the past two decades, but the intrinsic instability of liquid electrolytes results in safety issues. Solid polymer electrolytes would be a perfect solution to those safety issues, miniaturization and enhancement of energy density. However, as in liquids, the fraction of charge carried by lithium ions is small (<20%), limiting the power performances. Solid polymer electrolytes operate at 80 degrees C, resulting in poor mechanical properties and a limited electrochemical stability window. Here we describe a multifunctional single-ion polymer electrolyte based on polyanionic block copolymers comprising polystyrene segments. It overcomes most of the above limitations, with a lithium-ion transport number close to unity, excellent mechanical properties and an electrochemical stability window spanning 5V versus Li+/Li. A prototype battery using this polyelectrolyte outperforms a conventional battery based on a polymer electrolyte.