set theory – Is the leftmost branch function $text{UB}rightarrow omega^omega$ Borel?

Let’s denote by $text{UB}$ the set of ill-founded trees over $omega$ having a unique branch.
Consider now the function $b: text{UB} rightarrow omega^omega$, that associates to each tree $T$ in $text{UB}$ its unique branch $b_T$. Is this function Borel?

This function is the restriction of the left-most branch function defined over ill-founded trees (possibly with multiple branches), which is $sigma(boldsymbol{Sigma}_1^1)$-measurable, to $text{UB}$. Can we in some way prove that this restriction is indeed Borel? Otherwise, do we a sort of counterexample?