This is not really an answer, just remarks on the problem, but it would be too long for a comment. Apologies.

As remarked by François Dorais in a comment, you can phrase the question entirely in terms of polish spaces (i.e. completely metrizable and separable topological spaces). And since a connected metric space with at least two points is infinite, you are asking if there exists an infinite connected polish space such that any proper closed subspace is totally disconnected (equivalently, any proper closed connected subset is a point, or empty).

A first (failed) attempt to an example would be the *complete Erdös space* $E_c$ (Erdös, Annals of Math vol 41 1940), defined as the subspace of $\ell^2(\mathbb{N})$ where all coordinates are irrationals. It is polish and totally disconnected, but admits a *connectification* namely a (still polish) topology on $E_c\cup\{\infty\}$ that makes it connected (and of course induces the one on $E_c$). So it is rather "thinly" connected, but maybe not in your sense.

Another amazing property of $E_c$ is that it is homeomorphic to the subspace of $\ell^2(\mathbb{N})$ where all coordinates belong to $\{0\} \cup \{1/n\}_{n\geq 1}$. This sounds rather improbable.

All this (and much more) is explained in papers on JJ Dijkstra publications page
e.g. 27, 30 and 32.

Classical Descriptive Set Theory. $\endgroup$2more comments