Replacing the flammable liquid electrolyte currently used in most rechargeable lithium-ion batteries with a solid polymer electrolyte promises improved operational safety and increased energy density, e.g. by enabling lithium metal anodes. Polymer electrolytes typically suffer from low lithium-ion conductivity and limited electrochemical stability. We introduce a novel electrolyte based on a chemically cross-linked polysiloxane elastomer, modified with 3-mercaptopropiononitrile groups. The polysiloxane chains ensure high elasticity and low glass transition temperature, while the nitrile groups offer high dielectric permittivity and weak interaction with Li+. Combining these two properties into a solid polymer electrolyte results in excellent elasticity with no hysteresis after cyclic deformation, a low glass transition temperature (-51 degrees C), a high thermal stability up to at least 300 degrees C, an ionic conductivity of 4.8 x 10(-5) S cm(-1) at 60 degrees C, and a high transference number of 0.53. In all-solid-state symmetric lithium cells, this electrolyte enables stable lithium plating and stripping at 0.1 mA cm(-2) for over 1600 h at 60 degrees C. An all-solid-state full cell with a lithium iron phosphate cathode (areal capacity of 0.6 mA h cm(-2)) and lithium metal anode shows a high initial capacity of 134 mA h g(-1) and 75% capacity retention after 150 cycles at 0.1 mA cm(-2) at 60 degrees C. Preliminary results show that a room-temperature ionic conductivity as high as 6.4 x 10(-4) S cm(-1) and stable lithium plating and stripping at 0.2 mA cm(-2) for over 120 h at 22 degrees C can be achieved when the electrolyte is soaked in 1,2-dimethoxyethane.