I am trying to find an example of a scheme $X$ so that the morphism $Frob_X = (f, f^\flat)\colon X \to X$ given by $f = 1_x$ and $f^\flat\colon \Gamma(U, \mathscr{O}_X) \to \Gamma(U, \mathscr{O}_X), a \mapsto a^p$ induces an isomorphism on $\Gamma(X, \mathscr{O}_X)$ but is not a scheme automorphism.
I know that $X$ will have to be a scheme of characteristic $p.$ Also, $\Gamma(X, \mathscr{O}_x)$ must be isomorphic to $\mathbb{F}_p$ to guarantee that $f^\flat$ is an isomorphsim on the global sections. However, the obvious $X = \text{Spec} \mathbb{F}_p$ doesn't seem to work, as $Frob_X$ is an isomorphism in this case (the only open subsets of $\text{Spec} \mathbb{F}_p$ are itself and the empty set). I suspect I will have to glue affine schemes in some way to find such an $X$, but due to inexperience I can't seem to find a good place to start. Could someone provide a hint to point me in the right direction?