The set of regular points in an algebraic group

Question

I saw the following fact in a paper I was reading and I was wondering if someone could provide a reference. Let $K$ be a non-archimedean local field (say of characteristic $0$), and let $G$ be a connected reductive algebraic group over $K$. Let $G_r(K)$ be the set of $K$-regular points in $G(K)$. Then $G_r(K)$ is an open dense set in $G(K)$ (with respect to the topology induced by $K$). I would also be interested in determining whether this also true when $K$ has positive characteristic.