Existence of Connected Punctiform Space

Theorem
There exists at least one example of a connected topological space which is also punctiform.

Proof
Let $T$ be Cantor's leaky tent.

From Cantor's Leaky Tent is Connected, $T$ is a connected topological space.

From Cantor's Leaky Tent is Punctiform, $T$ is a punctiform space.

Hence the result.