On the $p$-adic transcendence measure of the values of functions satisfying some functional equations

Let $f(z)=f(z_1,\dots,z_8)$ be a transcendental function given by its Taylor series with coefficients from an algebraic field of finite degree over $Q$ in some neighbourhood $U$ of zero and satisfying Mahler type functional equations. Under some conditions on the function and on the algebraic point $\alpha=(\alpha_1,\dots,\alpha_8)\in U$ we compute the $p$-adic transcendence measure of $f(\alpha)$, $p$ being a prime number.