Anion-exchange-membrane fuel cells (AEMFCs) are a cost-effective alternative to proton-exchange-membrane fuel cells (PEMFCs). The development of high-performance and durable AEMFCs requires highly conductive and robust anion-exchange membranes (AEMs). However, AEMs generally exhibit a trade-off between conductivity and dimensional stability. Here, a fluorination strategy to create a phase-separated morphological structure in poly(aryl piperidinium) AEMs is reported. The highly hydrophobic perfluoroalkyl side chains augment phase separation to construct interconnected hydrophilic channels for anion transport. As a result, these fluorinated PAP (FPAP) AEMs simultaneously possess high conductivity (>150 mS cm(-1) at 80 degrees C) and high dimensional stability (swelling ratio <20% at 80 degrees C), excellent mechanical properties (tensile strength >80 MPa and elongation at break >40%) and chemical stability (>2000 h in 3 m KOH at 80 degrees C). AEMFCs with a non-precious Co-Mn spinel cathode using the present FPAP AEMs achieve an outstanding peak power density of 1.31 W cm(-2). The AEMs remain stable over 500 h of fuel cell operation at a constant current density of 0.2 A cm(-2).