Flexible integration of perovskite materials has driven diverse applications, from wearable detectors, portable energy systems to foldable displays. However, due to the intrinsic brittleness of perovskite, mechanical strain inevitably causes the degradation and variation of electronic performance of the devices. Here, we establish a periodic multilayered polymer-perovskite membrane that showcases plastic-like mechanical behaviors of small Young's modulus (5.41 GPa) and bending tolerance (radius of 0.5 mm), yet retains the perovskite's carrier transport capacity (μτ product of 1.04 × 10 cm V). The mechanistic study shows that the formation of bicontinuous perovskite-polyimide structure in the membrane accounts for the carrier transport and load transfer functions, respectively, thus unifies paradoxical mechanical and electronic properties. Using a lateral device configuration, X-ray detector based on the membrane delivers a high X-ray sensitivity of 8380.80 μC Gy cm, and withstands 30,000 repeated bending cycles under a bending radius of 1.5 mm without notable performance degradation.