Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map $$ J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty $$ returning the infinite jet at 0, which is a surjection by Borel's lemma. Here $\mathbb{R}^\infty$ is the set of all sequences of real numbers, with the family of seminorms induced by the truncations to the first $n$-coordinates. The map $J$ does not have a continuous linear section, but it has a continuous non-linear section. What I'd like to know is if $J$ has a smooth nonlinear section, even if just in a neighbourhood of $0$. Here smoothness is taken in the sense of Michal–Bastiani.

I recently learned that smooth maps $\mathbb{R}^n \to \mathbb{R}^\infty$ lift smoothly to $C^\infty(\mathbb{R},\mathbb{R})$ (via Enxin Wu's paper Homological Algebra for Diffeological Vector Spaces), which makes this projection maps a subduction of the associated diffeological spaces, making our friend $J$ above a diffeological principal bundle (which, I remind you, are not assumed locally trivial!). But I do wonder if it's a bundle in the traditional sense, in the category of Fréchet manifolds. Perhaps a continuous section could be smoothed, but I really am grasping at straws.