Let $V$ be a $\mathbb{Q}_p$-vector space endowed with a Hausdorff topology such that addition and scalar multiplication are continuous. Is $V$ necessarily totally disconnected?
I am in particular interested to the case of normed (or even Banach) $\mathbb{Q}_p$-vector spaces, with the topology induced by the norm. I know that ultrametric spaces are automatically totally disconnected but I'm not sure any normed vector space over $\mathbb{Q}_p$ is automatically ultrametric...