Let $p$ be a prime number, and let $\mathbb{Z}_p$ be the $p$-adic numbers. In concrete terms, what is the blow up of $\mathbb{A}^1_{\mathbb{Z}_p} = \operatorname{Spec} \mathbb{Z}_p[x]$ along the closed subscheme corresponding to the ideal $(p,x)$?
Edit: For context, this question comes up in the study of Néron models, but few authors bother to spell out all the details. Since my training in classical algebraic geometry is somewhat rudimentary it's not entirely clear to me how the construction goes. I looked in some standard references on algebraic geometry, but most of the examples are in the geometric context where one works over a field, and I don't immediately see how to convert these computations to my setting.
I think the answer to this question should not be difficult, but I also could not find any similar examples worked out in detail (although I may not have looked hard enough). It might be helpful to more people than myself to have the answer available here.
I actually wondered a similar question a while ago in the context of integral models of $\mathbb P^1$, so here are my thoughts.
The blow-up of a scheme $X$ by a sheaf of ideals $\mathcal I$, by definition, is the relative Proj $\underline{\mathrm{Proj}}_X(\bigoplus_{n\ge0}\mathcal I^n)$. Thus your blow up is $$\mathrm{Proj}(\mathbb Z_p[x]\oplus(p,x)\oplus(p^2,px,x^2)\oplus\cdots).$$ The ring has generators $X_1:=p$ and $Y_1:=x$ in degree $1$ with relations $xX_1=pY_1$. Thus the blow-up is isomorphic to the sub-scheme of $\mathbb A_{\mathbb Z_p}^1\times\mathbb P_{\mathbb Z_p}^1$, with coordinates $(x,(X_1:Y_1))$ cut out by $xX_1=pY_1$.
The special fiber over $p$ is the subscheme of $$\mathbb A^1_{\mathbb F_p}\times\mathbb P_{\mathbb F_p}^1$$ cut out by the equation $xX_1=0$, i.e., it is the union $\mathbb A^1\cup\mathbb P^1_{\mathbb F_p}$ where $0\in\mathbb A^1_{\mathbb F_p}$ and $(0:1)\in\mathbb P^1_{\mathbb F_p}$ is identified.
On the other hand, the generic fiber is the subscheme of $$\mathbb A^1_{\mathbb Q_p}\times\mathbb P^1_{\mathbb Q_p}$$ defined by $xX_1=pY_1$. Here, there is an isomorphism $$\begin{align*} \mathbb A^1_{\mathbb Q_p}&\xrightarrow\sim Z(xX_1-pY_1)\\ a&\mapsto(a,(p:a)). \end{align*}$$