  Let $K$ be an algebraically closed field of characteristic 0 that is complete
with respect to a non-archimedean absolute value. We establish a locally
uniform approximation formula of the Lyapunov exponent of a rational map $f$ of
$\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of
$n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and
$K$. As an immediate consequence, we obtain an estimate for the blow-up of the
Lyapunov exponent near a pole in one-dimensional families of rational maps over
$K$. Combined with our former archimedean version, this non-archimedean
quantitative approximation allows us to show:
  - a quantified version of Silverman's and Ingram's recent comparison between
the critical height and any ample height on the moduli space
$\mathcal{M}_d(\bar{\mathbb{Q}})$,
  - two improvements of McMullen's finiteness of the multiplier maps: reduction
to multipliers of cycles of exact given period and an effective bound from
below on the period,
  - a characterization of non-affine isotrivial rational maps defined over the
function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of
the growth of the degree of the multipliers.
