I want to know if connected reductive groups over non archimedean local fields have a dense countable subset. I was thinking that this should be true because if $G(\mathbb{F})$ is such group where $\mathbb{F}$ is a non archimedian local field then there exist an embedding of $G(\mathbb{F})$ into some matrix group GL$_n(\bar{\mathbb{F}})$. I then suppose that this map is continuous and open, right? and since $\bar{\mathbb{F}}$ has a dense countable set, or not? then $G(\mathbb{F})$ has a dense countable set. Can somone give me a reference of this result if it is true?

Thank you