Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ and $\gamma\in T(F)$, can we write $\gamma=exp(g)$ for some element $g\in\mathfrak{g}(F)$?

We can assume at first that $F$ is of characteristic zero and then for general characteristics is $p$ good sufficient?.