It suffices to find a space with the fixed point property such that $\mathrm{Cont}(X,X)$ has multiple connected components: Take a representative in each component and permute.

I think that $\mathbb{CP}^2$ has the fixed property and the space of self maps has multiple connected components. I hope I am not mistaken but I think that $[\mathbb{CP}^2,\mathbb{CP}^2]\cong [\mathbb{CP}^2,\mathbb{CP}^\infty]\cong \mathbb{Z}$. At least it is not too hard to find two self maps which are not homotopic: The constant map and the identity. Map all maps that are homotopic to the constant map to the identity and all maps that are not homotopic to the constant map to the constant map.

EDIT: I think you can prove very generally that $\mathrm{Cont(X,X)}$ does not have the fixed point property if $X$ is manifold of positive dimension. The space of continuous maps admits a model as a seperable Banach manifold. But any seperable Banach manifold of continuous maps is homeomorphic to a seperable Hilbert manifold $X$ (this is a theorem of Henderson (?)). Any separable infinite dimensional Hilbert manifold $Y$ is homeomorphic to $Y\times \mathbb{R}$. But for this you can write down a nonzero translation.