It is known that in the case of a Regular Language $L$ , the pumping lemma can be extended to apply to any sufficiently long subword of the language, ie, if $uwv \in L$ and $|w| \ge p$ then we can perform the usual breakup of $w$ as $w=xyz$ such that $uxy^izv \in L$.
Can we similarly extend the pumping lemma for context-free languages such that it is applicable to any sufficiently long subword? Intuitively I think it should generalize, since the pumping lemma is basically stems from the existence of a repeating non-terminal on some path given a sufficiently deep parse tree, and that should not change if we consider a subword instead of the entire word. But I'm having trouble with formalizing an argument.
If anyone could provide an idea for a formal argument or proof, it would be really appreciated.