Starting from this section till the end of the paper (except of the Section 11.7) we will assume the following
Zero Charactersistic Assumption. $K$ is a complete non-archimedean local field such that its residue field has characteristic zero.

Let $X$ be a $K$-analytic manifold of dimension $n$ and we are given a continuous map $\pi:X\to B$, where $B$ is a topological space. Then $B^{sm}$ carries a ${\bf Z}$-affine structure (Theorem 1). Suppose that there is an open $K$-analytic submanifold $U\subset X$ such that $\pi^{-1}(B^{sm})\subset U$ and there is a nonwhere vanishing analytic form $\Omega\in\Gamma(U,\Omega_{X}^{n})$. We are going to define a ${\bf Z}$-affine function $Val(\Omega)$ similarly to the definition of the function $Val(\varphi)$ in Section 4.1. Namely, in local coordinates $(z_{1},...,z_{n})$ we consider the expression $\varphi:=\Omega/\bigwedge_{1\leq i\leq n}(dz_{i}/z_{i})$. This is an invertible function, and we define $Val(\Omega)$ as $Val(\varphi)$. The independence on the choice of coordinates follows from the following lemma

Proof: By Lemma 1 from Section 4.1 we know that $z_{i}^{\prime}$ as any invertible function can be written in form $c_{i}z^{I^{(i)}}(1+o(1))$ for some nonzero $c_{i}\in K$ and a multi-index $I^{(i)}\in{\bf Z}^{n}$. Vectors $I^{(1)},\dots,I^{(n)}$ form a basis of ${\bf Z}^{n}$, as follows from the condition that $z_{1}^{\prime},\dots,z_{n}^{\prime}$ form a coordinate system. Therefore, after applying the change of coordinates $z_{i}\mapsto c_{i}z^{I^{(i)}}$ preserving form $\bigwedge_{i}dz_{i}/z_{i}$ up to sign, we may assume that $z_{i}^{\prime}=(1+o(1))z_{i}$. The Jacobian matrix of the transformation $(z_{i})\to(z_{i}^{\prime})$ is the identity matrix plus terms of size $o(1)$. Therefore its determinant has norm equal to 1. $\blacksquare$

Now we make the following
Constant Norm Assumption. The function $Val(\varphi)$ is locally constant.

Proof. Let us write in local coordinates $\Omega=\varphi(z_{1},...z_{n})\bigwedge_{1\leq i\leq n}{dz_{i}\over z_{i}}$. Define residue $Res(\Omega)\in K$ as the constant term $\varphi_{0}$ in the Laurent expansion $\varphi(z_{1},...z_{n})=\sum_{I\in{\bf Z}^{n}}\varphi_{I}z^{I}$. It is easy to see that $Res(\Omega)$ does not depend (up to a sign) on the choice of local coordinates. For non-vanishing everywhere $\Omega$ satisfying Constant Norm Assumption we have $\exp(-Val(\varphi))=|\varphi|=|\varphi_{0}|$. Therefore we have $Res(\Omega)\neq 0$.

Let us return to the proof of the Theorem. Let $F$ be the sheaf of abelian groups $F\subset\pi_{\ast}({\cal O}_{X}^{\times})$ consisting of $f$ such that $Val(f)=0$. Then we have an exact sequence of sheaves

where ${\cal O}_{K}$ denotes the constant sheaf with the fiber being the ring of integers of $K$. Indeed we embed $K^{\times}/{\cal O}_{K}^{\times}$ into $\pi_{\ast}({\cal O}_{X}^{\times})/F$ as constant functions. The projection $\pi_{\ast}({\cal O}_{X}^{\times})/F\to(T_{X}^{\ast})^{{\bf Z}}$ assigns to the function $f$ the linear part of the corresponding ${\bf Z}$-affine function $Val(f)$.

Notice that if $U\subset B^{sm}$ is a connected domain then any $f\in\Gamma(U,F)$ can be written (non-canonically) as $f=a(1+r)$, where $a\in{\cal O}_{K}^{\times}$ and $r=o(1)$ in $\pi^{-1}(U)$.

We define an epimorphism of sheaves $p_{\Omega}:F\twoheadrightarrow{\cal O}_{K}^{\times}$ by formula

Here $\exp$ and $\log$ are understood as infinite convergent series (in order to make sense of them we use Zero Characteristic Assumption).

It is easy to see that $p_{\Omega}$ is well-defined. Then the exact sequence of sheaves

defines a $K$-affine structure on $B^{sm}$ compatible with $Aff_{{{\bf Z}},B^{sm}}^{can}$. This concludes proof of the Theorem. $\blacksquare$

Notice that the above proof gives an explicit construction of the $K$-affine structure. We will denote it by $Aff_{K,B^{sm}}^{\Omega}$. It is easy to see that this $K$-affine structure does not change if we make a rescaling $\Omega\mapsto c\Omega,c\in K^{\times}$.