Recently i learned that we can't extend inverse function theorem in Euclidean space to manifold with boundary as domain. More precisely, if we have a smooth map $F : M \to N$ where $M$ is a manifold with boundary and $N$ is a manifold without boundary, for any $p \in M$ such that $dF_p$ invertible we can't always has a diffeomorphism $F|_{U_0} : U_0 \to V_0$ on the smaller nbhds $U_0$ of $p$ and $V_0$ of $F(p)$.
The example for such map is the inclusion map $\iota : \mathbb{H}^n \hookrightarrow \mathbb{R}^n$ see here or Lee's smooth manifold. I'm trying to prove for the general case but i'm fail. So i guess that there are map (with some manifold $M$ as domain) that we can extend IFT ? The problem is that i can't find an example for such map.
Is there any class of map (or class of manifold with boundary) such that we can extend IFT ?
Any help or hints will be appreciated. Thank you.