Is there an example of a smooth $n$manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$?
(This question was inspired by Trivial fiber bundle.)
Is there an example of a smooth $n$manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$? (This question was inspired by Trivial fiber bundle.) 

