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](https://ncatlab.org/nlab/show/Borel%27s+theorem). 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](https://ncatlab.org/nlab/show/Michal-Bastiani+smooth+map).

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](https://arxiv.org/abs/1406.6717)_), which makes this projection maps a _subduction_ of the [associated diffeological spaces](https://ncatlab.org/nlab/show/diffeological+space#RelationBetweenDeffeologicalAndFrechetStructure), 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.