Can every map between manifolds be factored as $p\circ i$

Question

Can every map between topological manifolds $f\colon X\to Y$ be factored as $p\circ i\colon X\to \overline{Y}\to Y$ with $i$ inclusion of open subset in another topological manifold $\overline{Y}$ and $p$ a proper map?(If topological manifold is not good enough, we may assume smooth manifolds)

Answer 1

No, consider for instance the universal cover $R\to S^1$.