Restriction of a covering map on a closed set having finite fibers is a proper map?

Question

Let $p\colon X\to Y$ be a covering map between two connected topological manifolds. Let $C$ be a closed and path-connected subset of $X$ such that each fiber of the restriction map $p\big |C\to Y$ is finite. Is $p\big|C\to Y$ a proper map?

I am trying to show $p$ is a closed map so that I can use this: A continuous closed map defined on a Hausdorff space having compact fibers is a proper map.

Also, note that a finite-fold covering map is a closed map.

Answer 1

Hint: This is false even for the universal covering $p$ of the once-punctured plane and $C$ such that $p|_C$ is 1-1.