Let $X$ be a contractible compact [edit: locally connected] topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?

No. I believe the first counterexample is from: Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 9698 which I unfortunately can't find online. Kinoshita's example is described on page 127 in this excellent article by Bing, however. EDIT: The question has been revised to add the local connectivity condition; as stated, I think the question is open. If "contractible" is replaced with "acyclic" there are counterexamples dating back to Borsuk, referenced e.g. here; Borsuk's paper, which I can't find online, is: K. Borsuk, Sur un continua acyclique qui se laisse transformer topologiquement en luimeme sans points invariants, Fund. Math. 24 (1935), pp. 5158. This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks. 

