  Lyman-$\alpha$ 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 ($M_{\star} = 5\times10^8$ 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 $\sim 25\times30$ 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.
