= Solution
\b[There is a <cofinal branch>.] We give a proof that does not require distinct limit-level nodes to have different predecessor chains.
The set $S=\{\alpha<\lambda:\operatorname{cf}(\alpha)=\kappa\}$ is stationary. Indeed a strictly increasing continuous $\kappa$-sequence in any <club set> has supremum in that <club set>, below $\lambda$, of <cofinality> $\kappa$. At each $\alpha\in S$, there are fewer than $\kappa$ pairs of nodes on $T_\alpha$. For every pair whose predecessor chains below $\alpha$ differ, choose a height where they differ. Since $\operatorname{cf}(\alpha)=\kappa$, all these heights are bounded by some $\beta_\alpha<\alpha$. Consequently nodes in $T_\alpha$ having the same predecessor at $\beta_\alpha$ have identical predecessor chains below $\alpha$.
The <Fodor lemma> states that a regressive <function> on a stationary <subset> of a regular uncountable <cardinal> is constant on a stationary <subset>. Applying it here, $\beta_\alpha$ has a constant value $\beta$ on a stationary <subset> $S'$. Choose $t_\alpha\in T_\alpha$ for each $\alpha\in S'$. The level $T_\beta$ has fewer than $\kappa<\lambda$ nodes. Partitioning $S'$ according to the predecessor of $t_\alpha$ at height $\beta$, one fiber $S''$ is stationary, since the union of fewer than $\lambda$ nonstationary sets is nonstationary. Let its common predecessor be $t$.
For $\alpha<\alpha'$ in $S''$, the predecessor of $t_{\alpha'}$ at level $\alpha$ and $t_\alpha$ have the same predecessor $t$ at level $\beta$. Their chains below $\alpha$ therefore agree. For each $\nu<\lambda$, choose $\alpha\in S''$ above $\nu$ and let $b_\nu$ be the predecessor of $t_\alpha$ at level $\nu$. The preceding comparison makes $b_\nu$ independent of that choice. The nodes $b_\nu$ form a chain through every level:
$$
\boxed{\{b_\nu:\nu<\lambda\}\text{ is a cofinal branch of }T.}
$$
This proves the <uniformly narrow regular-height tree branch theorem>. The uniform bound below the smaller regular $\kappa$ is stronger than merely bounding each level below $\lambda$.
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