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Direct and mixing-induced $C\!P$-violating asymmetries in $B^0_s \to K^+K^-$ decays are measured for the first time using a data sample of $pp$ collisions, corresponding to an integrated luminosity of $1.0~\mathrm{fb}^{-1}$, collected with the LHCb detector at a centre-of-mass energy of $7~\mathrm{TeV}$. The results are $C_{KK} = 0.14 \pm 0.11 \pm 0.03$ and $S_{KK} = 0.30 \pm 0.12 \pm 0.04$, where the first uncertainties are statistical and the second systematic. The corresponding quantities are also determined for $B^0 \to \pi^+\pi^-$ decays to be $C_{\pi\pi} = -0.38 \pm 0.15 \pm 0.02$ and $S_{\pi\pi} = -0.71 \pm 0.13 \pm 0.02$, in good agreement with existing measurements