  Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive
polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P
\in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in
\mathbb{N}} (\deg{F^{n}})^{1/n}$ the dynamic degree of $F$ and $\alpha(P) :=
\limsup_{n \in \mathbb{N}} h_K(F^{n}P)^{1/n}$ the arithmetic degree of $F$ at
$P$. We have $\alpha(P) \leq \delta$, and extending a conjecture of Silverman
from the number field case, it is expected that equality holds if the orbit of
$P$ is Zariski-dense.
  We prove a weaker form of this conjecture: if $\delta > 1$ and the orbit of
$P$ is Zariski-dense, then also $\alpha(P) > 1$. We obtain furthermore a more
precise result concerning the growth along the orbit of $P$ of the heights of
the individual coordinates, and formulate a few related open problems motivated
by our results, including a generalization "with moving targets" of Faltings's
theorem back in the number field case.
