We present the first detailed abundance analysis of the metal-poor giant HK-II 17435-00532. This star was observed as part of the University of Texas Long-Term "Chemical Abundances of Stars in the Halo" (CASH) Project. A spectrum was obtained with the High Resolution Spectrograph (HRS) on the Hobby-Eberly Telescope with a resolving power of R 15,000. Our analysis reveals that this star may be located on the red giant branch, red horizontal branch, or early asymptotic giant branch. We find that this metal-poor ([Fe/H]=-2.2) star has an unusually high lithium abundance (log ɛ (Li)=+2.1), mild carbon ([C/Fe]=+0.7) and sodium ([Na/Fe]=+0.6) enhancement, as well as enhancement of both s-process ([Ba/Fe]=+0.8) and r-process ([Eu/Fe]=+0.5) material. The high Li abundance can be explained by self-enrichment through extra mixing mechanisms that connect the convective envelope with the outer regions of the H-burning shell. If so, HK-II 17435-00532 is the most metal-poor star in which this short-lived phase of Li enrichment has been observed. The r- and s-process material was not produced in this star but was either present in the gas from which HK-II 17435-00532 formed or was transferred to it from a more massive binary companion. Despite the current non-detection of radial velocity variations (over a time span of 180 days), it is possible that HK-II 17435-00532 is in a long-period binary system, similar to other stars with both r and s enrichment. We acknowledge support from the W.J. McDonald Fellowship of McDonald Observatory (to A.F), NASA's AAS Small Research Grant Program and the GALEX GI grant 05-GALEX05-27 (to J.R.), the Italian MIUR-PRIN06 Project "Late phases of Stellar Evolution: Nucleosynthesis in Supernovae, AGB Stars, Planetary Nebulae" (to R.G.), and the U.S. National Science Foundation (grants AST06-07708 to C.S., AST04-06784, AST07-07776 and PHY02-15783 to T.C.B., and AST 07-07447 to J.J.C.).