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Every fibration with totally path-disconnected fibers has the unique path lifting property (2.2.5 of E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966) implying that $p_{\ast}$ is injective. However, it seems to me that $p_{\ast}$ is not likely to be surjective unless you impose more conditions on the spaces $Y$, $\widetilde{Y}$, or the map $p$. In particular, for every based map $f:X\to Y$, there is a unique based function $\tilde{f}:X\to\widetilde{Y}$ such that $p\circ\tilde{f}=f$ (defined using the fact that every path has a unique lift). I doubt that $\tilde{f}$ is continuous in general.