We present a first sample of 117 [O iii] λλ4960, 5008–selected star-forming galaxies at 5.33 < z < 6.93 detected in JWST/NIRCam 3.5 μm slitless spectroscopy of a
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field centered on the hyperluminous quasar SDSS J0100+2802, obtained as part of the Emission-line galaxies and Intergalactic Gas in the Epoch of Reionization (EIGER) survey. Three prominent galaxy overdensities are observed, one of them at the redshift of the quasar. Galaxies are found within 200 pkpc and 105 km s−1 of four known metal absorption-line systems. We focus on the role of the galaxies in ionizing the intergalactic medium (IGM) during the later stages of cosmic reionization and construct the mean Lyα and Lyβ transmission as a function of distance from the galaxies. At the lowest redshifts in our study, 5.3 < z < 5.7, the IGM transmission rises monotonically with distance from the galaxies, as seen previously at lower redshifts. In contrast, at 5.7 < z < 6.14, the transmission of both Lyα and Lyβ first increases with distance but then peaks at a distance of 5 cMpc before declining. Finally, in the region 6.15 < z < 6.26, where the additional ionizing radiation from the quasar dominates, the monotonic increase in transmission with distance is reestablished. This result is interpreted to represent evidence that the transmission of the IGM at z ∼ 5.9 toward J0100+2802 results from the “local” ionizing radiation of galaxies that dominates over the much-reduced cosmic background.