In Professor Lee's Introduction to Smooth Manifolds, 2nd edition, in the section at the end of Chapter 5 on Submanifolds with Boundary, he states (just before Proposition 5.49 on page 121)
Many (though not all) of the earlier results in this chapter have analogues for submanifolds with boundary.
He then goes on to state a Proposition and two Theorems which are of that type. But absent is any mention of whether or not the Extension Lemma for Functions on Submanifolds (Lemma 5.34) holds in some form for submanifolds with boundary, nor are submanifolds with boundary mentioned in the associated Problem 5-18, which has to do with converses of Lemma 5.34.
My question is whether or not there are versions of Lemma 5.34 and Problem 5-18 that apply to submanifolds with boundary. If so, are they just simple restatements with submanifold replaced by submanifold with boundary, or is there something more needed?