The version of Zermelo's fixed-point theorem I have in mind is the following (from Davey and Priestley, Theorem 8.23):
Let $P$ be a dcppo (a pointed, directedly complete partial order) and let $F$ be an increasing (i.e. $x \leq F(x)$ for all $x \in P$) self-map on $P$. Then $F$ has a minimal fixpoint.
In Exercise 8.20 they outline a proof, in which one considers the minimal $F$-invariant sub-dcppo $P_0$ of $P$. Using minimality one shows that every element $x \in P_0$ is a roof, which means that if $y < x$, then $F(y) \leq x$ (it is not clear whether $y$ must lie in $P_0$, not sure if it matters), and that if $x,y \in P_0$ then either $y \leq x$ or $F(x) \leq y$. From this one establishes that $P_0$ is a chain, and hence has a maximum $\top_{P_0}$. This is clearly a fixed-point of $F$, but I am having trouble showing that it is a minimal fixed-point.
How do I show that $\top_{P_0}$ is indeed minimal?
(Just to avoid potential confusion: Note that $\top_{P_0}$ need not be the least fixed-point. Also note that $F$ is not necessarily monotone, in which case this would be easy!)
I now believe this to be false. Consider the poset $P = (\omega^\partial)_\bot$, i.e. the set $\mathbb{N} \cup \{\bot\}$ with the ordering $\preceq$ given by
$$ \bot \prec \cdots \prec n \prec n-1 \prec \cdots \prec 1 \prec 0. $$
This is clearly a dcppo (indeed, every nonempty subset even has a maximum) and define $F \colon P \to P$ by $F(\bot) = 0$ and $F(n) = n$ for $n \in \mathbb{N}$. Then $F$ is clearly increasing in the above sense, but every natural number is a fixed-point of $F$, so it has no minimal fixed-point.