We employ high-speed video imaging to study microdroplets of a few picoliters volume impacting freely suspended smectic liquid-crystal film. Depending on the impact parameters, in particular, droplet velocity and mass, three different regimes are observed such as trapping, rebounding, and tunneling. Fast droplets penetrate the films completely. After they have passed the film, they are coated with a layer of film material while the original smectic film remains intact. Droplets in a certain intermediate velocity range bounce back from the film. After impact, the film deforms and hurls the droplet back, depleting a substantial share of the initial kinetic energy. Slow droplets are caught and embedded in the film. During impact and tunneling, an appreciable amount of kinetic energy is lost. The energy is partially dissipated during droplet impact and during subsequent mechanical vibrations and oscillations of the film and the droplet. The tunneling process of high-speed droplets can be exploited to prepare smectic shells of well-defined sizes that enclose picoliters of an immiscible liquid.