Smectic liquid crystals are fluids, and in most rheological situations they behave as such. Nevertheless, when thin freely floating films of smectic A or smectic C materials are compressed quickly in-plane, they resist such stress by buckling similar to solid membranes under lateral stress. We report experimental observations of wrinkling and bulging of finite domains within the films, so-called islands, and give a qualitative explanation of different observed patterns. Depending on the external stress and their dimensions, the islands can expel a specifically shaped bulge in their center, form radial wrinkles or develop target-like wrinkle structures. When the external stress is relaxed, these patterns disappear reversibly.