Outstanding Li+ conductivity and diffusivity have been achieved in free-standing ion gel electrolytes synthesized by in-situ photopolymerization of 1-(2-methacryloyloxy)ethyl-3-butylimidazolium bis(trifluoromethane sulfonyl) imide (IMMA) and/or poly(ethylene glycol) methacrylate (EGMA), in the presence of the room temperature ionic liquids 1-ethyl-3-methylimidazolium bis(fluorosulfonyl)imide (EMIFSI), 1-butyl-1-methylpyrrolidinium bis(fluorosulfonyl) imide (BMPFSI) and bis(trifluoromethane)sulfonimide lithium salt (LiTFSI). The membranes are easy to handle and thermally stable up to 200 degrees C. Those containing IMMA in the polymer chain present liquid-like ionic conductivities (up to 10 mS cm(-1) at 25 degrees C), and liquid-like Li+ diffusivities and conductivities (D-Li approximate to 4 x 10(-11) m(2) s(-1), sigma(Li)approximate to 1.4 mS cm(-1) at 25 degrees C) unreported so far in a solid electrolyte. D-Li is not only very high but significantly higher than its counteranions' diffusivity, D-FSI or D-TFSI, a very rare behavior in electrolytes where transport is, in principle, ruled by viscosity. It is proposed that in these polycationic electrolytes the motion of Li+ occurs via two different transport mechanisms, the well-known viscosity-governed transport and an additional anion-exchange mechanism that enables very fast Li+ diffusion. This combination has high practical relevance for Li+ batteries as it implies a high contribution of sigma(Li) to the overall electrolyte's conductivity, and it constitutes a breakthrough in the design of polymer-based solid electrolytes for Li.