Let $X$ be a smooth projective variety over a scheme $S$ being the spectrum of a discrete valuation ring of mixed characteristic $(0,p)$. Let $X_n$ be the respective thickenings of the reduced special fiber $X_1$. Then there is the following algebraization isomorphism: $$Pic(X)\xrightarrow{\sim} \varprojlim Pic(X_n).$$

How can one deduce this isomorphism from Grothendieck's formal existence theorem [EGA3, 5.1.4], saying that for a coherent $\mathcal{O}_X$-module $\mathcal{F}$, there is an isomorphism $$H^i(X,\mathcal{F})\rightarrow H^i(\hat{X},\hat{\mathcal{F}}),$$ where $\hat{X}$ is the formal scheme obtained by completing $X$ along $X_1$?

(The first idea that comes to my mind is considering $H^1$ and the module $\mathcal{O}_X^*$ - which might not be coherent? - but I don't understand the object on the right).

What are the problems if I want to consider $Pic(X_1)\otimes \mathbb{Z}/p\mathbb{Z}$, i.e. $p$ being the characteristic of the residue field of the base? Why does one have to consider thickenings instead? (this question could be put in a more general context, i.e. different cohomology theories mod $p$ in char $p$).