Although two-dimensional conjugated metal-organic frameworks (2D c-MOFs) provide an ideal platform for precise tailoring of capacitive electrode materials, high-capacitance 2D c-MOFs for non-aqueous supercapacitors remain to be further explored. Herein, we report a novel phthalocyanine-based nickel-bis(dithiolene) (NiS₄)-linked 2D c-MOF (denoted as Ni₂[CuPcS₈]) with outstanding pseudocapacitive properties in 1 M TEABF₄/acetonitrile. Each NiS₄ linkage is disclosed to reversibly accommodate two electrons, conferring the Ni₂[CuPcS₈] electrode a two-step Faradic reaction with a record-high specific capacitance among the reported 2D c-MOFs in non-aqueous electrolytes (312 F g^-1) and remarkable cycling stability (93.5% after 10,000 cycles). Multiple analyses unveil that the unique electron-storage capability of Ni₂[CuPcS₈] originates from its localized lowest unoccupied molecular orbital (LUMO) over the nickel-bis(dithiolene) linkage, which allows the efficient delocalization of the injected electrons throughout the conjugated linkage units without inducing apparent bonding stress. The Ni₂[CuPcS₈] anode is used to demonstrate an asymmetric supercapacitor device that delivers a high operating voltage of 2.3 V, a maximum energy density of 57.4 Wh kg^-1, and ultralong stability over 5000 cycles.