Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) defined over $k$. Is it true that the restriction of $f$ to $k$ rational points $k^n\to V(k)$ surjective?

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) defined over $k$. Is it true that the restriction of $f$ to $k$ rational points $k^n\to V(k)$ surjective?

After it is answered I realized that I simplified what I want to know too much. Please see the comments for the answer for more information.