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We present the first microlensing candidate for a free-floatingexoplanet-exomoon system, MOA-2011-BLG-262, with a primary lens mass of M_host~ 4 Jupiter masses hosting a sub-Earth mass moon. The data are well fit by thisexomoon model, but an alternate star+planet model fits the data almost as well.Nevertheless, these results indicate the potential of microlensing to detectexomoons, albeit ones that are different from the giant planet moons in oursolar system. The argument for an exomoon hinges on the system being relativelyclose to the Sun. The data constrain the product M pi_rel, where M is the lenssystem mass and pi_rel is the lens-source relative parallax. If the lens systemis nearby (large pi_rel), then M is small (a few Jupiter masses) and thecompanion is a sub-Earth-mass exomoon. The best-fit solution has a largelens-source relative proper motion, mu_rel = 19.6 +- 1.6 mas/yr, which wouldrule out a distant lens system unless the source star has an unusually highproper motion. However, data from the OGLE collaboration nearly rule out a highsource proper motion, so the exoplanet+exomoon model is the favoredinterpretation for the best fit model. However, the alternate solution has alower proper motion, which is compatible with a distant (so stellar) host. ABayesian analysis does not favor the exoplanet+exomoon interpretation, soOccam's razor favors a lens system in the bulge with host and companion massesof M_host = 0.12 (+0.19 -0.06) M_solar and m_comp = 18 (+28 -100 M_earth, at aprojected separation of a_perp ~ 0.84 AU. The existence of this degeneracy isan unlucky accident, so current microlensing experiments are in principlesensitive to exomoons. In some circumstances, it will be possible todefinitively establish the low mass of such lens systems through themicrolensing parallax effect. Future experiments will be sensitive to lessextreme exomoons.

A Sub-Earth-Mass Moon Orbiting a Gas Giant Primary or a High Velocity Planetary System in the Galactic Bulge