= Solution
For a regular uncountable <kappa-tree>, keep exactly the nodes whose extensions have unbounded heights:
$$
T^*=\{t\in T:\sup\{\operatorname{ht}(u):u\ge_Tt\}=\kappa\}.
$$
This <set> is predecessor-closed. At any level $\alpha$, if no node survived, the extension heights above each of its fewer than $\kappa$ nodes would be bounded. Regularity would give a single bound for their union, contradicting the height of the original <set-theoretic tree>. Thus every level of $T^*$ is nonempty and still has size less than $\kappa$.
If $t\in T^*$ and $\beta>\operatorname{ht}(t)$, consider its extensions at level $\beta$. If none survived, fewer than $\kappa$ bounded extension <sets> would again bound every extension of $t$, a contradiction. Therefore a surviving level-$\beta$ extension exists. Hence
$$
\boxed{T^*\text{ is a well-pruned }\kappa\text{-subtree}.}
$$
This <unbounded-extension kernel of a regular tree> uses regularity essentially. If one permits singular-height <set-theoretic trees> in the term “$\kappa$-tree”, the unrestricted assertion is false: take fewer than $\kappa$ disjoint branches with lengths cofinal in a singular $\kappa$. The levels are small and the height is $\kappa$, but no node has unbounded extensions. A common root can be added without creating a well-pruned subtree. The usual regular-height convention is therefore the one used here.
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