The mechanical response of a smectic A liquid single crystal elastomer (LSCE) to stress applied perpendicular and parallel to the layers is described. We find that the static modulus perpendicular to the layers is two orders of magnitude larger than the modulus parallel to the layer direction, where the latter is completely due to the crosslinking. Macroscopically the width of the smectic A LSCE does not shrink when a mechanical stress is applied parallel to the layers while the length of the film increases by a factor of 2 thus demonstrating the in-plane fluidity of the smectic layering.