Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose that the closed unit ball of $X$ is compact in the strong topology.
Question 1.
Is $X$ necessarily isomorphic to some finite dimensional Banach space ?
Question 2. If the answer for the question 1 is no , can we always find in $\mathcal{L}(X,X)$ an unbounded linear operator ?
Different from this question Is there an infinite-dimensional Banach space with a compact unit ball? I would like to assume the axiom of choice.
Comment. The example I have in mind is $(\mathbb{R}^n,\|\cdot\|_{2})$ as $\mathbb{Q}$ vector space. Clearly $\text{dim}_{\mathbb{Q}}\ \mathbb{R}^n=\infty$ and the unit ball is compact and this space is complete with respect to the standard Euclidean norm $\|\cdot\|_2$, but in this example both questions 1 and 2 are trivial.