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?



The generalization of Brouwer's fixed point theorem?
No. I believe the first counterexample is from:


Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96-98


which I unfortunately can't find online. Kinoshita's example is described on page 127 in this excellent [article by Bing](http://www.jstor.org/stable/2317258), 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](http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBkQFjAA&url=http%253A%252F%252Fmatwbn.icm.edu.pl%252Fksiazki%252Ffm%252Ffm107%252Ffm10711.pdf&ei=VteSTYmMDInmsQPd_PW9Cw&usg=AFQjCNH6ceZLPpJxw5lVy0sc-p_8DdudaQ&sig2=Zr9Sn7SIVPRC1dpyhggXAA); Borsuk's paper, which I can't find online, is:


K. Borsuk, Sur un continua acyclique qui se laisse transformer topologiquement en lui-meme sans points invariants, Fund. Math. 24 (1935), pp. 51-58.


This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks. 


