Let $X$ be a Banach space and fix a convex compact set $K\subseteq X$ with $0\notin K$.
Is it true that there exists $x \in X$ such that $ \|x\|>\max_{k \in K} \|x-k\|? $
The answer is positive in the case $X=\mathbf{R}^k$ (and, more generally, in Hilbert spaces): draw a "suitable" line passing through the origin which intersects $K$ and choose a point $x$ which is sufficiently far away from $0$, on the side of $K$ (see details below).
What about the general case in Banach spaces? (A previous version of the question was in the context of locally convex topological vector spaces with a compatible translation-invariant metric; cf. comments of user8268.)