With the number of confirmed rocky exoplanets increasing steadily, their characterization and the search for exoplanetary biospheres are becoming increasingly urgent issues in astrobiology. To date, most efforts have concentrated on the study of exoplanetary atmospheres. Instead, we aim to investigate the possibility of characterizing an exoplanet (in terms of habitability, geology, presence of life, etc.) by studying material ejected from the surface during an impact event. For a number of impact scenarios, we estimate the escaping mass and assess its subsequent collisional evolution in a circumstellar orbit, assuming a Sun-like host star. We calculate the fractional luminosity of the dust as a function of time after the impact event and study its detectability with current and future instrumentation. We consider the possibility to constrain the dust composition, giving information on the geology or the presence of a biosphere. As examples, we investigate whether calcite, silica, or ejected microorganisms could be detected. For a 20km diameter impactor, we find that the dust mass escaping the exoplanet is roughly comparable to the zodiacal dust, depending on the exoplanet's size. The collisional evolution is best modeled by considering two independent dust populations, a spalled population consisting of nonmelted ejecta evolving on timescales of millions of years, and dust recondensed from melt or vapor evolving on much shorter timescales. While the presence of dust can potentially be inferred with current telescopes, studying its composition requires advanced instrumentation not yet available. The direct detection of biological matter turns out to be extremely challenging. Despite considerable difficulties (small dust masses, noise such as exozodiacal dust, etc.), studying dusty material ejected from an exoplanetary surface might become an interesting complement to atmospheric studies in the future.