Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric.

Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to V$ are equivalent via **smooth** homotopy?

edited: (for any k)