Ly \alpha photons are resonantly scattered by neutral hydrogen , and may therefore trace both the spatial extent and the kinematics of the gas surrounding galaxies . We present new observations of the extended Ly \alpha halo of Q2343-BX418 , a low mass ( M _ { \star } = 5 \times 10 ^ { 8 } M _ { \odot } ) , low metallicity ( Z \approx 0.25 Z _ { \odot } ) star-forming galaxy at z = 2.3 . Using the Keck Cosmic Web Imager , the blue-sensitive optical integral field spectrograph recently installed on the Keck II telescope , we detect Ly \alpha in emission to a radius of 23 kpc , and measure an exponential scale length of 6 kpc in the outer region of the extended halo . We study the double-peaked spectroscopic Ly \alpha profile in individual spectral pixels ( “ spaxels ” ) over a \sim 25 \times 30 kpc region , finding significant variations in the peak ratio and peak separation . The profile is dominated by the red peak in the central regions , while in the outskirts of the extended halo the red and blue peak strengths are roughly equal ; these observations are consistent with a model in which the peak ratio is largely determined by the radial component of the outflow velocity . We find a gradient of 300 km s ^ { -1 } in the Ly \alpha peak separation across the extended halo , indicating variations in the column density , covering fraction or velocity range of the gas . These new observations emphasize the need for realistic , spatially resolved models of Ly \alpha radiative transfer in the halos of galaxies .