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.

On the other hand, if you only mean "foliation" as in your title, and not "fibration", then there is Vogt's foliation of R^3 by circles! (But it is not C^1, only differentiable). Vogt, Elmar, "A foliation of R3 and other punctured 3-manifolds by circles", Publications Mathématiques de l'IHÉS, Tome 69 (1989), p. 215-232 http://www.numdam.org/item/PMIHES_1989__69__215_0/