  We present deep Keck/MOSFIRE $H$-band spectroscopic observations covering the
[CIII],CIII]$\lambda\lambda1907,1909$ doublet for three $z\sim8$ galaxy
candidates in the AEGIS field. Along with non-detections in two galaxies, we
obtain one of the highest-redshift detections to-date of [CIII]$\lambda 1907$
for the galaxy AEGIS-33376, finding $z_{\rm spec}=7.945\pm0.001$. We measure a
[CIII]$\lambda$1907 flux of $2.24\pm0.71\times10^{-18} \mbox{ erg}\mbox{
s}^{-1} \mbox{ cm}^{-2}$, corresponding to a rest-frame equivalent width of
$20.3\pm6.5 \unicode{x212B}$ for the single line. Given the not very
constraining upper limit for CIII]$\lambda 1909$ based on strong sky-line
contamination, we assume a [CIII]$\lambda$1907/CIII]$\lambda 1909$ doublet
ratio of 1.5 and infer a total [CIII],CIII]$\lambda\lambda1907,1909$ equivalent
width of $33.7\pm 10.8 \unicode{x212B}$. We repeat the same reductions and
analysis on multiple subsets of our raw data divided on the basis of time and
observing conditions, verifying that the [CIII]$\lambda 1907$ emission is
present for AEGIS-33376 throughout our observations. We also confirm that the
significance of the [CIII]$\lambda 1907$ detection in different subsets of our
data tracks that of brighter emission features detected on the same multi-slit
mask. These multiple tests suggest that the observed emission line is real and
associated with the $z\sim 8$ target. The strong observed
[CIII],CIII]$\lambda\lambda1907,1909$ in AEGIS-33376 likely indicates ISM
conditions of low metallicity, high ionization parameter, and a hard ionizing
spectrum, although AGN contributions are possible. This single detection
represents a sizable increase in the current sample
[CIII],CIII]$\lambda\lambda1907,1909$ detections at $z>7$, while
$\textit{JWST}$ will provide the first statistical samples of such measurements
at these redshifts.
