Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not a point.



Foliation of $\mathbb R^n$ by connected compact manifolds
There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See


Borel, Armand; Serre, Jean-Pierre,
Impossibilité de fibrer un espace euclidien par des fibres compactes,
C. R. Acad. Sci. Paris 230 (1950), 2258–2260. 


