In [Con99, Definition 2.2.2] Conrad defined the notion of irreducibility for analytic spaces which we recall here. Let $X$ be a paracompact strictly $K$-analytic space and $p:\tilde{X}\rightarrow X$ the normalization of $X$ ([Con99, 2.1]). Then the irreducible components of $X$ are defined to be the sets $X_{i}:=p(\tilde{X}_{i})$ where $\tilde{X}_{i}$ are the connected components of $\tilde{X}$. The space $X$ is said to be irreducible if it has a unique irreducible component. By [Con99, Lemma 2.2.3] $X$ is irreducible if and only if it can not non trivially be written as a union of two closed strictly $K$-analytic subsets.
Let $Y$ be an irreducible component of $X$ and $V=\mathscr{M}(\mathscr{A})$ an affinoid domain with $Y\cap V\neq\emptyset$. Then by [Con99, Corollary 2.2.9] there is an irreducible component $Y^{\prime}$ of $V$ which is contained in $V\cap Y$. Then $Y^{\prime}$ corresponds to a minimal prime ideal $\mathfrak{p}$ of $\mathscr{A}$ and hence to an irreducible component of $\Spec(\mathscr{A})$. We define the multiplicity of $Y$ to be the multiplicity of this component. Note that this does not depend on the choice of $V$ and $Y^{\prime}$: If $V^{\prime}=\mathscr{M}(\mathscr{B})\subseteq V$ and $\mathfrak{p}^{\prime}$ is a minimal prime ideal of $\mathscr{B}$ lying over $\mathfrak{p}$ then $\mathscr{B}/\mathfrak{p}\mathscr{B}$ is reduced by [BGR84, Corollary 7.3.2/10] as it induces an affinoid domain in $\mathscr{M}(\mathscr{A}/\mathfrak{p})$ which is reduced. Hence also $\mathscr{B}_{\mathfrak{p}^{\prime}}/\mathfrak{p}\mathscr{B}_{\mathfrak{p}^{\prime}}$ is reduced and since $\mathscr{B}_{\mathfrak{p}^{\prime}}$ is a local ring of dimension 0, this implies $\mathfrak{p}^{\prime}\mathscr{B}_{\mathfrak{p}^{\prime}}=\mathfrak{p}\mathscr{B}_{\mathfrak{p}^{\prime}}$. Hence by [Ful98, Lemma A.4.1] the multiplicity of the irreducible component corresponding to $\mathfrak{p}$ is equal to that of the irreducible component corresponding to $\mathfrak{p}^{\prime}$.
Let $\varphi:X\rightarrow Y$ be a proper surjective morphism of irreducible and reduced strictly $K$-analytic spaces. If $\dim(Y)<\dim(X)$ we set $\deg(\varphi)=0$. Otherwise $\varphi$ is a finite morphism outside a lower dimensional analytic subset $W$ of $Y$. Let $\mathscr{M}(\mathscr{A}^{\prime})$ be an affinoid domain in $Y\setminus W$, $V$ an irreducible component of $\Spec(\mathscr{A}^{\prime})$ and $\mathscr{M}(\mathscr{A}):=\varphi^{-1}(\mathscr{M}(\mathscr{A}^{\prime}))$ then $\Spec(\mathscr{A})\rightarrow\Spec(\mathscr{A}^{\prime})$ is finite and we define $\deg(\varphi)$ to be the sum of the degrees of the irreducible components of $\Spec(\mathscr{A})$ over $V$. As explained in [Gub98, 2.6] this again does not depend on the choices.