Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map).
Can we say that $f$ is a proper map, i.e. pre images of compact subsets of $V$ are compact in $U$? What if $X$ is compact?
If not, I'd also be interested in a counterexample.