There are many forms of Lagrange inversion. The ones that don't involve division by integers are valid in positive characteristic. For example: Given a power series $R(t)$, there is a unique power series $f=f(x)$ such that $f(x) = x R(f(x))$, and for any Laurent series $\phi(t)$ and $\psi(t)$ and any integer $n$ we have
$$[x^n]\phi(f)=[t^n]\bigl(1-tR'(t)/R(t)\bigr)\phi(t)R(t)^n$$ and $$[x^n]\frac{\psi(f)}{ 1-xR'(f)}=[t^n]\psi(t)R(t)^n.$$