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We propose and demonstrate both flexible and stretchable blue light-emitting diodes based on core/shell InGaN/GaN quantum well microwires embedded in polydimethylsiloxane membranes with strain-insensitive transparent electrodes involving single-walled carbon nanotubes. InGaN/GaN core-shell microwires were grown by metal-organic vapor phase epitaxy, encapsulated into a polydimethylsiloxane film, and then released from the growth substrate. The fabricated free-standing membrane of light-emitting diodes with contacts of single-walled carbon nanotube films can stand up to 20% stretching while maintaining efficient operation. Membrane-based LEDs show less than 15% degradation of electroluminescence intensity after 20 cycles of stretching thus opening an avenue for highly deformable inorganic devices.