We present deep Keck/MOSFIRE $H$-band spectroscopic observations covering the [CIII],CIII]$lambdalambda1907,1909$ doublet for three $zsim8$ 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 $z_{rm spec}=7.945pm0.001$. We measure a [CIII]$lambda$1907 flux of $2.24pm0.71times10^{-18} mbox{ erg}mbox{ s}^{-1} mbox{ cm}^{-2}$, corresponding to a rest-frame equivalent width of $20.3pm6.5 unicode{x212B}$ 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]$lambdalambda1907,1909$ equivalent width of $33.7pm 10.8 unicode{x212B}$. 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 $zsim 8$ target. The strong observed [CIII],CIII]$lambdalambda1907,1909$ 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]$lambdalambda1907,1909$ detections at $z>7$, while $textit{JWST}$ will provide the first statistical samples of such measurements at these redshifts.
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