Nanostructured ultraviolet (UV) light sources represent a growing research field in view of their potential applications in wearable optoelectronics or medical treatment devices. In this work, we report the demonstration of the first flexible UV-A light emitting diode (LED) based on AlGaN/GaN core-shell microwires. The device is based on a composite microwire/poly(dimethylsiloxane) (PDMS) membrane with flexible transparent electrodes. The electrode transparency in the UV range is optimized: namely, we demonstrate that single-walled carbon nanotube electrodes provide a stable electrical contact to the membrane with high transparency (70% at 350 nm). The flexible UV-A membrane demonstrating electroluminescence around 345 nm is further applied to excite Zn-Ir-BipyPDMS luminophores: the UV-A LED is combined with the elastic luminophore-containing membrane to produce a visible amber emission from 520 to 650 nm. The obtained results pave the way for flexible inorganic light-emitting diodes to be employed in sensing, detection of fluorescent labels, or light therapy.