= Solution
Use a compatible complete metric on the <Polish space> $P$. For a <tree on a Polish space> $T\subseteq\bigcup_nP^n$, let
$$
r_T(t)=\sup\{r_T(t^\frown x)+1:t^\frown x\in T\}
$$
be its <rank of a well-founded tree>, with terminal rank zero. We bound this rank using a countable approximation tree, rather than assuming that the original tree has countably many nodes.
Choose a countable base of open balls. A node of the approximation tree $S$ at depth $n$ records, for every coordinate $i\leq j\leq n$, a basic ball $U_{j,i}$. Require its diameter to be less than $2^{-j}$, require $\overline{U_{j+1,i}}\subseteq U_{j,i}$, and require
$$
T_j\cap\prod_{i=1}^jU_{j,i}\ne\varnothing\qquad(1\leq j\leq n).
$$
Initial truncation removes the last row of balls. There are only countably many such finite records, so $S$ is countable.
An infinite branch in $S$ would give, by completeness and shrinking nested balls, a limit $p_i\in P$ for each coordinate. For fixed $j$, witnesses from later levels of $T$ have their first $j$ coordinates converging to $(p_1,\ldots,p_j)$. Initial-segment closure puts those witnesses in $T_j$, and closedness of $T_j$ puts the limit there too. Hence $(p_i)$ would be an infinite branch of $T$. Thus $S$ is well-founded.
Every tuple $t=(p_1,\ldots,p_n)\in T_n$ can be fitted inside such a record with each $p_i\in U_{n,i}$. Every extension $t^\frown p$ can be fitted in a child record: shrink the old balls around their actual points and choose a small ball around the new point. A well-founded induction therefore gives $r_T(t)\leq r_S(s)$ whenever the record $s$ fits $t$. At the empty record this bounds the whole tree's rank by $r_S(\varnothing)$.
The rank of a countable well-founded tree is countable: at each node it is a supremum of countably many successors of countable ordinals, by well-founded recursion. Consequently \b[a closed well-founded tree on a Polish space has countable height]. The completeness and closedness assumptions are precisely what turn a branch of approximation records into an actual branch.
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