Microporous sheet product and methods of making and using the same. In one embodiment, the microporous sheet product is made by a process that includes melt-extruding a sheet material using an extrusion mixture that includes (i) a cyclic olefin copolymer, (ii) an electrolyte swellable thermoplastic, and (iii) a compatibilizing agent that promotes mixing of the cyclic olefin copolymer and the electrolyte swellable thermoplastic, the compatibilizing agent having a boiling point in the range of 135-300° C. As an example, the cyclic olefin copolymer may be an ethylene-norbornene copolymer, the electrolyte swellable thermoplastic may be polyethylene oxide, and the compatibilizing agent may be mineral spirits. After extrusion, the sheet material may be cooled, and the compatibilizing agent may be removed, forming an ionically-conductive microporous sheet product. The microporous sheet product has high-temperature stability and gels when exposed to a liquid electrolyte, enabling high ionic conductivity when used as a battery separator.