Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$. Examples to keep in mind: $(K,R,k) = (\mathbb{Q}_p, \mathbb{Z}_p, \mathbb{F}_p)$ or $(\mathbb{C}((t)), \mathbb{C}[[t]], \mathbb{C})$ or $(\mathbb{C}_p, \mathcal{O}_{\mathbb{C}_p}, \bar{\mathbb{F}}_p)$.
Let $\mathcal{O}\langle T_1,\dots,T_d\rangle = \{\sum_I a_I T^I : |a_I| \rightarrow 0 \text{ as } |I| \rightarrow \infty \}$ be the Tate algebra with integral coefficients and let $F = (F_1,\dots,F_n)$ be an $n$-tuple of elements in $\mathcal{O}\langle T_1,\dots,T_d\rangle$. Evaluation of power series defines a map $F\colon \mathcal{O}^d \rightarrow \mathcal{O}^n$.
I have a few basic questions concerning the image $X:=F(\mathcal{O}^d)\subset \mathcal{O}^n$ which seem to fall under the rubric of rigid analytic spaces but I couldn't find an obvious reference for it.
Question 1: Is $X$ contained in a closed rigid analytic space of $\mathcal{O}^n$ of dimension $\leq d$?
Next we consider the natural map $\mathbb{P}\colon \mathcal{O}^n\setminus \{0\} \rightarrow \mathbb{P}^{n-1}(K)$ given by projectivizing coordinates: $(x_1, \dots,x_n) \mapsto (x_1 : \dots : x_n)$. Let $Y\subset \mathbb{P}^{n-1}(K)$ be the image of $X\setminus \{0\}$ under $\mathbb{P}$.
Let $Z\subset \mathbb{P}^{n-1}(k)$ be the image of $Y$ under the reduction map $\mathbb{P}^{n-1}(K) = \mathbb{P}^{n-1}(\mathcal{O}) \rightarrow \mathbb{P}^{n-1}(k)$.
Question 2: Is $Z$ contained in a closed algebraic subvariety of $\mathbb{P}^{n-1}_k$ of dimension $\leq d$?
Note that the image of $X$ under the reduction map $\mathcal{O}^n \rightarrow k^n$ is obviously algebraic, since the reduction mod $\mathfrak{m}$ of the $F_i$ are polynomials. However this is not obvious for $Z$, as far as I can see. Does it follow from the theory of formal models?
Any help would be appreciated!
