Recent theoretical models suggest that the early phase of galaxy formation could involve an epoch when galaxies are gas rich but inefficient at forming stars: a "dark galaxy" phase. Here, we report the results of our Multi-Unit Spectroscopic Explorer (MUSE) survey for dark galaxies fluorescently illuminated by quasars at z>3. Compared to previous studies which are based on deep narrowband (NB) imaging, our integral field survey provides a nearly uniform sensitivity coverage over a large volume in redshift space around the quasars as well as full spectral information at each location. Thanks to these unique features, we are able to build control samples at large redshift distances from the quasars using the same data taken under the same conditions. By comparing the rest-frame equivalent width (EW0) distributions of the Ly{alpha} sources detected in proximity to the quasars and in control samples, we detect a clear correlation between the locations of high-EW0 objects and the quasars. This correlation is not seen in other properties, such as Ly{alpha} luminosities or volume overdensities, suggesting the possible fluorescent nature of at least some of these objects. Among these, we find six sources without continuum counterparts and EW0 limits larger than 240{AA} that are the best candidates for dark galaxies in our survey at z>3.5. The volume densities and properties, including inferred gas masses and star formation efficiencies, of these dark galaxy candidates are similar to those of previously detected candidates at z~2.4 in NB surveys. Moreover, if the most distant of these are fluorescently illuminated by the quasar, our results also provide a lower limit of t=60Myr on the quasar lifetime.