If one is doing smooth representations of the $p$-adic group $G$ on complex vector spaces,
then the action is locally constant (small open subgroups act trivially on vectors),
and so any interpretation of what it means to differentiate the action will give you
the trivial action. (Note also the the Lie algebra has $p$-adic coefficients, while
the representation $V$ has complex coefficients, so there is not much scope for a non-trivial action!).

On the other hand, if you mean not-necessarily-smooth actions of $G$ on a $p$-adic vector space, as one has in the theory developed by Schneider and Teitelbaum that is related to the $p$-adic Langlands program, then one can and does differentiate the action (provided it
is locally analytic), and one does obtain interesting Lie algebra actions.