We can always get non-principal ultrafilters on sets from non-discrete extremally disconnected spaces.
Let $X$ be a topological space. Let $\text{Clo}(X)$ denote the Boolean algebra of clopen subsets of $X$. If $X$ is extremally disconnected and zero dimensional, then $\text{Clo}(X)$ is a complete Boolean algebra, and for each non-isolated point $x_0\in X$, the set $\{R\in\text{Clo}(X):x_0\in R\}$ is a non-principal ultrafilter.
If $B$ is a complete Boolean algebra, and $U$ is an ultrafilter on $B$, then for each partition $p$ of $B$, we can define an ultrafilter $U_p=\{R\subseteq p:\bigvee R\in U\}$ on the set $p$.
Proposition: Suppose that $\kappa$ is a regular cardinal. Let $B$ be a $<\kappa$-complete Boolean algebra. Then an ultrafilter $U$ on $B$ is $<\kappa$-complete if and only if whenever $p$ is a partition of $B$ with $|p|<\kappa$, the ultrafilter $U_p$ is principal.
As a consequence, if $B$ is a complete Boolean algebra, and $U$ is a non-principal ultrafilter on $B$, then there is some partition $p$ where $U_p$ is a non-principal ultrafilter on $p$.
In point-free topology, we have extremally disconnected frames, but in point-free topology, one does not need to worry about ultrafilters at all.