Let $k$ be an arbitrary field (in my case $k = \mathbb Q_p$), and $G\subset \mathrm{GL}(n)_{/k}$ a reductive group. Let $G^0$ be its identity connected component.

Suppose that $G^0$ contains an element with distinct eigenvalues (and hence a Zariski-dense subset of such elements).

Is it the case that every connected component of $G$ contains an element with distinct eigenvalues?