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...

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    $\begingroup$ In this generality, no. If you take V with the indiscrete topology, then addition and scalar multiplication are continuous for trivial reasons but V is connected, also for trivial reasons. $\endgroup$ – Asvin Jun 24 at 14:35
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    $\begingroup$ @HagenvonEitzen isn't a zero-dimensional v.s. trivial, thus totally disconnected for any topology? $\endgroup$ – frafour Jun 24 at 15:10
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    $\begingroup$ Not without further restrictions. $\mathbb C_p \simeq \mathbb C$ as fields (with the axiom of choice). Endow the $\mathbb Q_p$- vector space $\mathbb C_p$ with the usual archimedean topology of $\mathbb C$, which is a field topology, i.e. multiplication and addition are continuous. This topology is Hausdorff but not totally disconnected. $\endgroup$ – Torsten Schoeneberg Jun 24 at 15:22
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    $\begingroup$ @TorstenSchoeneberg Are you sure the scalar multiplication $\mathbb{Q}_p\times\mathbb{C}\to\mathbb{C}$ is continuous? $\endgroup$ – user10354138 Jun 24 at 15:41
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    $\begingroup$ @user10354138: Very good point. If one endows $\mathbb{Q}_p$ with the subspace topology, it is, but I see the OP probably asks for continuity of scalar multiplication with the $p$-adic topology on scalars; then this does not work. $\endgroup$ – Torsten Schoeneberg Jun 24 at 16:53

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