We fabricated [Co/Cu] multilayers revealing a giant magnetoresistance (GMR) effect on free-standing elastic poly(dimethylsiloxane) (PDMS) membranes. The GMR performance of [Co/Cu] multilayers on rigid silicon and on free-standing PDMS is similar and does not change with tensile deformations up to 4.5%. Mechanical deformations imposed on the sensor are totally reversible, due to the elasticity of the PDMS membranes. This remarkable performance upon stretching relies on a wrinkling of GMR layers on top of the PDMS membrane.