Liquid-crystalline polymers are materials of considerable scientific interest and technological value. An important subset of these materials exhibit rubber-like elasticity, combining the optical properties of liquid crystals with the mechanical properties of rubber. Moreover, they exhibit behaviour not seen in either type of material independently, and many of their properties depend crucially on the particular mesophase employed. Such stretchable liquid-crystalline polymers have previously been demonstrated in the nematic, chiral-nematic, and smectic mesophases. Here, we report the fabrication of a stretchable gel of blue phase I, which forms a self-assembled, three-dimensional photonic crystal that remains electro-optically switchable under a moderate applied voltage, and whose optical properties can be manipulated by an applied strain. We also find that, unlike its undistorted counterpart, a mechanically deformed blue phase exhibits a Pockels electro-optic effect, which sets out new theoretical challenges and possibilities for low-voltage electro-optic devices.