Let $ U$ be a connected unipotent algebraic group over a field of characteristic $ p>0$ . Assume $ U$ acts on an affine variety $ X$ by regular maps.

Is it true that the stabilizers of rational points $ x\in X$ under this action are connected?

Edit as the comments below point out, the answer is apparently- no, with an abundance of counterexamples.

Some comments The question obviously has a positive answer in the case of characteristic $ 0$ (since the stabilizer is also unipotent alg. gp.). Also, in the case where $ X$ is a vector space and $ U$ acts on $ X$ by linear transformations, one can also show that all stabilizer in $ U$ are connected. Lastly, if $ p$ is large enough so that the exponential map is defined on $ U$ , it seems plausible that one can apply it in order to obtain that the stabilizer of elements in $ X$ are connected. However, this is an assumption I would avoid if possible.

I think I have an argument for the general case. However, the argument seems to be rather elaborate, and I’m not completely confident that I haven’t missed anything, so I thought it might be wise to ask if anyone here is aware of a counter example, or a reference to the proof.

One last comment while I cannot assume that $ p$ is larger than some huge number (therefore, I wish to avoid applying the exponential map), it is possible to omit ‘small’ subsets of primes. In particular, it is possible to assume $ p$ is ‘good’ in some sense.

Thank you!