K can reversibly intercalate into graphite by forming KC8 (279 mA h g-1, ≈0.2 V) in conventional carbonate electrolytes, but the large ionic radius of K+ (1.38 Å) easily results in structural degradation and rapid capacity decay. Here, commercial graphite particles are directly electrospun into network-like carbon fibers, thus forming a flexible Gr@CNF membrane. This hybrid electrode configuration can efficiently withstand the volume expansion during K+ insertion. K||Gr@CNF half-cells can stably operate for over 800 cycles (running time of 170 days at C/3) and achieve fast K+-intercalation kinetics at 5C. The fabricated Gr@CNF||AC K-ion hybrid capacitor delivers a high energy density of 119.3 W h kg-1 at 2717.82 W kg-1, corresponding to a fast-charge time of 3.2 min.