Methylammonium lead iodide (MAPbI3) perovskite materials show an outstanding performance in photovoltaic devices. However, certain material properties, especially the possible ferroic behavior, remain unclear. We observed distinct nanoscale periodic domains in the piezoresponse of MAPbI3(Cl) grains. The structure and the orientation of these striped domains indicate ferroelasticity as their origin. By correlating vertical and lateral piezoresponse force microscopy experiments performed at different sample orientations with x-ray diffraction, the preferred domain orientation was suggested to be the a1-a2-phase. The observation of these ferroelastic fingerprints appears to strongly depend on the film texture and thus the preparation route. The formation of the ferroelastic twin domains could be induced by internal strain during the cubic-tetragonal phase transition.