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Soft deformable materials are needed for applications such as stretchable electronics, smart textiles, or soft biomedical devices. However, the design of a durable, cost-effective, or biologically compatible version of such a material remains challenging. Living animal cells routinely cope with extreme deformations by unfolding preformed membrane reservoirs available in the form of microvilli or membrane folds. We synthetically mimicked this behavior by creating nanofibrous liquid-infused tissues that spontaneously form similar reservoirs through capillarity-induced folding. By understanding the physics of membrane buckling within the liquid film, we developed proof-of-concept conformable chemical surface treatments and stretchable basic electronic circuits.