Covalent organic polymers (COPs) show great potential as supercapacitor electrodes due to their tunable properties, stability, and high capacitance. In this work, we prepared a covalent organic polymer (Py-DSDA-COP) through a Schiff base reaction of the 1,3,6,8-tetrakis(4-formylphenyl)pyrene monomer (Py-Ph-CHO) and 4,4′-disulfanediyldianiline (DSDA), validated its chemical structure, and investigated its thermal stability, porosity, morphology, and electrochemical properties. The Py-DSDA-COP displayed high semi-crystallinity, good porosity, and a specific capacitance of 56.18 F g−1. To improve its electrochemical performance, we incorporated different nano-carbonaceous materials [fullerene (C60), MWCNTs, and SWCNTs] into Py-DSDA-COP and compared their characteristics. The Py-DSDA-COP/SWCNTs showed the highest specific capacitance (171 F g−1) at 1 A g−1and highest energy density (23.7 W h kg−1) with a capacity retention of 93% after 2000 cycles, which is superior to those of Py-DSDA-COP/MWCNTs and Py-DSDA-COP/C60 nanocomposites. These superior electrochemical characteristics of Py-DSDA-COP/SWCNTs are attributed to the excellent π–π stacking interactions, higher EDLC behavior, higher conductivity, lower diameter, and higher surface area of SWCNTs than those of MWCNTs.