I believe that the following result is true, but am having a hard time finding a suitable reference to convince myself:
Suppose $X:\mathbb{R}^n\to \mathbb{R}^n$ is analytic. If the smooth function $u:(0,\infty)\times \mathbb{R}^n\to \mathbb{R}$ solves the parabolic equation
$$\frac{\partial u}{\partial t}=\Delta u+ X\cdot \nabla u,$$ then for each $t>0$, the mapping $x\mapsto u(t,x)$ is analytic.
The elliptic version of this result is frequently cited as "well-known" in the literature, and in fact, there is this result for elliptic non-linear systems as well: https://www.jstor.org/stable/2372830.
The parabolic result is for the regular heat equation ($X=0$) is discussed here: Space analyticity of solution to heat equation. However, the proof it refers to appears to lean quite heavily on an explicit expression for the heat kernel/fundamental solution, which might not be available if $X\neq 0$.
Thanks in advance.
I think the result you’re looking for is given in:
Classes of solutions of linear systems of partial differential equations of parabolic type by Avner Friedman.
Duke Math. J. 24 (3), 433-442, (September 1957) DOI: 10.1215/S0012-7094-57-02450-X