While I do not have an answer to your question, here are two related results:
- If $X$ is a 1-dimensional contractible continuum, then $X$ satisfies the FPP (every continuous self-map $X\to X$ has a fixed point). See:
Young, G. S., Fixed-point theorems for arcwise connected continua, Proc. Am. Math. Soc. 11, 880-884 (1961). ZBL0102.37806.
- Suppose that $X$ is an acyclic compact ANR. Then $X$ satisfies the FPP. See Corollary 8.10 on page 68 in
Górniewicz, L., On the Lefschetz fixed point theorem, Brown, R.F. (ed.) et al., Handbook of topological fixed point theory. Berlin: Springer (ISBN 1-4020-3221-8/hbk). 43-82 (2005). ZBL1077.55001.
In order to connect to your question, note that every ANR is locally contractible and almost conversely every finite dimensional metrizable locally contractible space is an ANR.
My guess is that one should assume in your question local contractibility instead of local connectivity, but I do not have an example.