Example of a local homeomorphism that does not have the path lifting property.

Question

I am looking for an example of a local homeomorphism $p:X \to Y$, that does not have the path lifting property.

I demand $X$ and $Y$ be path-connected, Hausdorff.

Note: I am asking this question out of curiosity. So I have no idea how difficult this might be.

Answer 1

Take $p : (0,2\pi +1) \to S^1, p(t) = e^{it}$. Now try to lift the path $u : I \to S^1, u(t) = e^{it}$.