Differences in the elemental abundances of planet-hosting stars in binary systems can give important clues and constraints about planet formation and evolution. In this study we performed a high-precision, differential elemental abundance analysis of a wide binary system, HD 80606/80607, based on high-resolution spectra with high signal-to-noise ratio obtained with Keck/HIRES. HD 80606 is known to host a giant planet with the mass of four Jupiters, but no planet has been detected around HD 80607 so far. We determined stellar parameters as well as abundances for 23 elements for these two stars with extremely high precision. Our main results are that (i) we confirmed that the two components share very similar chemical compositions, but HD 80606 is marginally more metal-rich than HD 80607, with an average difference of +0.013 ± 0.002 dex (σ = 0.009 dex); and (ii) there is no obvious trend between abundance differences and condensation temperature. Assuming that this binary formed from material with the same chemical composition, it is difficult to understand how giant planet formation could produce the present-day photospheric abundances of the elements we measure. We cannot exclude the possibility that HD 80606 might have accreted about 2.5 to 5 M_Earth material onto its surface, possibly from a planet destabilised by the known highly eccentric giant.