We have conducted a highly sensitive census of the evolved-star population in the metal-poor dwarf galaxy Leo P and detected four asymptotic giant branch (AGB) star candidates. Leo P is one of the best examples of a nearby analog of high-redshift galaxies because of its primitive metal content (2% of the solar value), proximity, and isolated nature, ensuring a less complicated history. Using medium-band optical photometry from the Hubble Space Telescope (HST), we have classified the AGB candidates by their chemical type. We have identified one oxygen-rich source which appears to be dusty in both the HST and Spitzer observations. Its brightness, however, suggests it may be a planetary nebula or post-AGB object. We have also identified three carbon-rich candidates, one of which may be dusty. Follow-up observations are needed to confirm the nature of these sources and to study the composition of any dust that they produce. If dust is confirmed, these stars would likely be among the most metal-poor examples of dust-producing stars known and will provide valuable insight into our understanding of dust formation at high redshift.