The question is fairly straightforward:
Is there a topological division ring $D$ of characteristic $p$ that is Hausdorff and connected?
I do not assume that inversion is continuous, that is, only addition and multiplication are assumed to be continuous under the topology in question.
Since the connected component of $0$ is an ideal, any topological division ring is either connected or totally disconnected. I cannot think of a single connected topological $\mathbb{F}_p$ algebra, but couldn't point down a proof that they are all totally disconnected either. I'm asking this here as I suppose this might be known.
In the paper:
the authors present for every characteristic a construction of a topological division ring $D$ of that characteristic that is Hausdorff and connected. Moreover, $D$ is actually commutative with continuous inversion (so a topological field) and path connected.
The precise construction is as follows: start with any integral domain $k$ and let $R$ be a polynomial ring over $k$ where one independent variable $T_\alpha$ is added to $k$ for every real $\alpha \in [0,1]$, and then kill the relations $T_0 = 0$, $T_1 = 1$. The authors define a norm $|\cdot|$ on $R$ using the norm on $(0,1)$ which induces a ring topology such that the map $(0,1) \to R$ that maps $\alpha$ to $T_\alpha$ is an isometry, from which follows that this ring is path connected. The norm is such that $|T_\alpha - T_\beta| = |\alpha - \beta|$.
This topology is later show to extend into a Hausdorff field topology on the field of fractions $K$ of $R$ such that the inclusion $R \to K$ is a homeomorphism onto its image. In particular, since $0$ and $1$ are connected by a path in $R$, they so are in $K$, which shows that $K$ is also path connected. It is clear that $K$ has the same characteristic as the initial integral domain $k$.