The polymer synthesis of 2,3,5,6-tetrafluoro-4-pyridinecarbonitrile-3,3,3′,3′-tetramethyl-1,1′-spirobisindane-5,5′,6,6′-tetrol copolymer, termed PIM-Py, was investigated under different solvent (dimethylformamide (DMF) and dimethylacetamide/dichlorobenzene) and temperature (65–160 °C) conditions to produce a range of topologically different polymer samples. Characterization of the polymers, particularly with proton NMR spectroscopy and multiple detector SEC analysis, indicated that, like PIM-1, the polymerizations proceeded with a degree of polymer chain branching. This is attributed to the occurrence of monosubstitution reactions, instead of disubstitution, which eventually leads to a significant proportion of colloidal network formation. However, all polymer samples remained soluble/dispersible in chloroform at the concentration required to cast self-standing films. This work reports the first examination of PIM-Py as a membrane for gas separation applications. The most structurally diverse PIM-Py samples produced films that exhibited selectivity/permeability balances in single gas permeation studies above the 2008 Robeson upper bound for the CO2/N2 gas pair. Indeed, a film cast from the highest colloidal network content sample surpassed the recently introduced 2019 CO2/N2 upper bound. After 143 days of aging, a 40 μm self-standing membrane still exhibited a single gas CO2 permeability of 4480 barrer and an ideal CO2/N2 selectivity of 45. The polymers produced in lower temperature reactions in DMF exhibited gas separation performances very similar to a structurally regular “normal” PIM-1 polymer, sitting on or around the 2008 Robeson upper bound line. Single gas permeation measurements to determine CO2/CH4 selectivity showed similar trends across the range of polymer samples, without generally reaching high selectivities as for the CO2/N2 pair. Mixed gas CO2/CH4 permeation measurements with aging were also completed for PIM-Py membranes, which indicated similar gas separation performance to a structurally regular PIM-1 polymer. This study would suggest that, like PIM-1, gas separation performance of PIM-Py is greatly influenced by the topological balance toward branched and network material within the polymer sample.