The Golomb space $(\mathbb N,\tau)$ (a "universal" counterexample to many questions), also gives a counterexample with $|\mathbb N|=\aleph_0$ and $|\tau|=\mathfrak c$.
I recall that the Golomb space is the set $\mathbb N$ of natural numbers endowed with the topology $\tau$ generated by the base consisting of the arithmetic progressions $a+\mathbb N_0b=\{a+nb:n\ge 0\}$ where $a,b$ are relatively prime.
It is well-known that the Golomb space is connected and Hausdorff. Since it contains a countable disjoint family of open sets (like any infinite Hausdorff space), its topology has cardinality $\mathfrak c\le|\tau|\le|\mathcal P(\mathbb N)|=\mathfrak c$.
In place of the Golomb space one can take any other countable Hausdorff connected space.
Such spaces have appeared in other questions of Dominic van der Zypen:
Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?
Is $\mathbb{Q}$ the continuous image of a Golomb-like space, or vice versa?
Cardinality of a set of countable connected Hausdorff spaces
Continuous self-maps in the Golomb space that are neither increasing nor decreasing