Single-ion conducting polymer electrolytes are considered particularly attractive for realizing high-performance solid-state lithium-metal batteries. Herein, a polysiloxane-based single-ion conductor (PSiO) is investigated. The synthesis is performed via a simple thiol-ene reaction, yielding flexible and self-standing polymer electrolyte membranes (PSiOM) when blended with poly(vinylidene fluoride-co-hexafluoropropylene) (PVdF-HFP). When incorporating 57 wt% of organic carbonates, these polymer membranes provide a Li$^{+}$ conductivity of >0.4 mS cm$^{-1}$ at 20 °C and a wide electrochemical stability window of more than 4.8 V. This excellent electrochemical stability allows for the highly reversible cycling of symmetric Li||Li cells as well as high-energy Li||LiNi$_{0.6}$Mn$_{0.2}$Co$_{0.2}$O$_{2}$2 (NMC$_{622}$) and Li||LiNi$_{0.8}$Mn$_{0.1}$Co$_{0.1}$O$_{2}$ (NMC$_{811}$) cells for several hundred cycles at relatively high discharge and charge rates. Remarkably, Li||NMC$_{811}$ cells with high mass loading cathodes provide more than 76% capacity retention at a high current density of 1.44 mA cm$^{-2}$, thus rendering this polymer electrolyte suitable for high-performance battery applications.