Let $K$ be an algebraically closed field, complete with respect to a nonarchimedean valuation. Chapter 2 of the book "Rigid Analytic Geometry and Its Applications" by Fresnel and Van der Put begins with the definition of affinoid subsets of the projective line $\mathbb P^1(K)=K$ $\cup$ {$\infty$}. The definitions are as follows:
An open disk $D \subset \mathbb P^1$ is a subset that has the form: {$z \in K : \vert z - a \vert < r$} with $a \in K$ and $r \in \vert K^* \vert $ or {$z \in K : \vert z - a \vert > r$} $ \cup $ {$\infty $}. We will for simplicity write {$z \in \mathbb P^1 : \vert z - a \vert < r$} or {$z \in \mathbb P^1 : \vert z - a \vert > r$}. The definition of a closed disk is obtained from the above by replacing < and > by $\leq$ and $\geq$. We note that the terminology is somewhat misleading since an open disk (and similarly a closed disk) is both open and closed in the ordinary topology of $\mathbb P^1$. A connected affinoid subset of $\mathbb P^1$ is the complement of a non-empty finite union of open disks. An aflinoid subset of $\mathbb P^1$ is a finite union of connected aflinoid subsets. The empty set is (in general) also considered an affinoid subset.
According to these definitions, the closed disk $B:=${$z \in \mathbb P^1 : \vert z \vert \leq 1$} is a connected affinoid (its complement being the open disk {$z \in \mathbb P^1 : \vert z \vert > 1$}) and $C:=${$z \in \mathbb P^1 : \vert z \vert \geq 1$} is a connected affinoid (its complement being the open disk {$z \in \mathbb P^1 : \vert z \vert < 1$}).
Hence $\mathbb P^1$ is an affinoid subset of itself, being the union of the two connected affinoids $B$ and $C$.
Now Chapter $2$ of said book is intended to build intuition for the more technical Chapters $3$ and $4$ on affinoid algebras and rigid spaces. But I assume that the terminology was chosen in such a way that the affinoid sets in the sense of Chapter 2 later produce affinoid spaces in the sense of Chapters $3$ and $4$ (i.e. in the sense of classical rigid analytic geometry). And indeed, Example $4.3.2$ in the book says that any affinoid $F$ subset of $\mathbb P ^1$ can be identified with the affinoid space $Sp(\mathcal O (F))$.
Does this mean that the rigid projective line $\mathbb P ^1$ is an affinoid rigid analytic space?
The answer probably being "no", why did Fresnel and Van der Put choose the call the sets defined in Chapter 2 "affinoid"?