An infinitesimal extension of an affine scheme $\operatorname{Spec}R$ is a surjection $\hat R\twoheadrightarrow R$ with nilpotent ideal. The scheme case is defined by globalizing.
I read somewhere on MSE that for a smooth $\Bbbk$-scheme the tangent bundle is again a scheme. The SDG picture of tangent spaces as infinitesimal vector spaces about a point lead me to wonder whether the tangent bundle of such a scheme is always an infinitesimal extension of it.
Is the tangent bundle of a smooth $\Bbbk$-scheme an infinitesimal extension of it?
No. Most basically, the arrow goes the wrong way: the tangent bundle of a space $X$ is a map $TX \to X$, so for affine schemes the corresponding map on rings of functions goes the other way.
Also, the tangent vector directions in the tangent bundle are not themselves infinitesimal directions. For example, the tangent bundle of the affine line is just the affine plane, another ordinary variety with no non-reduced behavior.