Let $K$ be a non
archimedean algebraically closed field of characteristic $\pi$,
complete for its ultrametric absolute value. In a recent paper by
Escassut and Yang (\cite{EY6}) polynomial decompositions
$P(f)=Q(g)$ for meromorphic functions $f$, $g$ on $K$ (resp. in a
disk $d(0, r^-)\subset K$) have been considered, and for a class
of polynomials $P$, $Q$, estimates for the Nevanlinna function
$T(\rho,f)$ have been derived.\\ In the present paper we consider
as a generalization rational decompositions of meromorphic
functions, i.e., we discuss properties of solutions $f$, $g$ of
the functional equation $P(f)=Q(g)$, where $P, \;Q$ are in $K(x)$
and satisfy a certain condition (M). We infer that in the case,
where $f$, $g$ are analytic functions, the Second Nevanlinna
Theorem yields an analogue result as in the mentioned paper
\cite{EY6}. However, if they are meromorphic, non trivial
estimates for $T(\rho,f)$ are more sophisticated.