What the correct notion of "Kähler differentials" on a sufficiently nice adic spaces (rigid space, perhaps) ? Given, a smooth variety $X$ over a perfect field $k$ of some positive characteristic $p$ and a formal lift $\mathcal{X} \to X$ from $k$ to $W(k)$, equipped with a lift of Frobenius $\sigma_{\mathcal{X}}$, then thanks to Bhatt, Lurie, and Matthew, one knows that the de Rham-Witt complex $W\Omega^*_{X/k}$ becomes the usualy de Rham complex of $X/k$ after reduction modulo $p$, i.e.:
$$W\Omega^*_X/p \cong_{qis} \Omega^*_{X/k}$$
and furthermore, that:
$$W\Omega^*_{\mathcal{X}/W(k)} \cong_{qis} \Omega^*_{\mathcal{X}/W(k)}$$
which means that:
$$\Omega^*_{\mathcal{X}/W(k)}/p \cong_{qis} \Omega^*_{X/k}$$
My initial guess was that one can obtain the de Rham complex $\Omega^*_{\mathcal{X}_{\eta}/W(k)}$ of the generic fibre $\eta$ of $\mathcal{X}/W(k)$ (where the pullback is taken in the category of adic spaces) by simply performing extension scalars to $W(k)[1/p]$ on $\Omega^*_{\mathcal{X}/W(k)}$, but then I got stuck when I tried figure out whether or not there is a non-discrete topology on $H^i(\Omega^*_{\mathcal{X}_{\eta}/W(k)[1/p]})$. Even if there is such a topology, I also have no idea how to prove the exactness of $\Omega^*_{\mathcal{X}_{\eta}/W(k)[1/p]}$.
By extension, how does one define connections on vector bundles on a given adic space ? If $\mathcal{E}$ is a vector bundle on $\mathcal{X}_{\eta}$, then will a connection simply be a homomorphism of $\mathcal{O}_{\mathcal{X}_{\eta}}$-modules:
$$\nabla: \mathcal{E} \otimes_{\mathcal{O}_{\mathcal{X}_{\eta}}} \mathcal{O}_{\mathcal{X}_{\eta}} \to \mathcal{E} \otimes_{\mathcal{O}_{\mathcal{X}_{\eta}}} \Omega^1_{\mathcal{X}_{\eta}/W(k)[1/p]}$$
or is there a subtlety I'm overlooking ?
