  Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of
$\mathbb{Q}_p$, $K$ the maximal unramified extension of $\mathbb{Q}_p$,
$\overline{K}$ some fixed algebraic closure of $K$, and $\mathbb{C}_p$ the
completion of $\overline{K}$. Let $G_F$ the absolute Galois group of $F$. Let
$A$ be an abelian variety defined over $F$, with good reduction. Classically,
the Fontaine integral was seen as a Hodge--Tate comparison morphism, i.e. as a
map $\varphi_{A} \otimes 1_{\mathbb{C}_p}\colon
T_p(A)\otimes_{\mathbb{Z}_p}\mathbb{C}_p\to
\text{Lie}(A)(F)\otimes_F\mathbb{C}_p(1)$, and as such it is surjective and has
a large kernel.
  The present article starts with the observation that if we do not tensor
$T_p(A)$ with $\mathbb{C}_p$, then the Fontaine integral is often injective. In
particular, it is proved that if $T_p(A)^{G_K} = 0$, then $\varphi_A$ is
injective. As an application, we extend the Fontaine integral to a perfectoid
like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$, then
$A(\overline{K})$ has a type of $p$-adic uniformization, which resembles the
classical complex uniformization.
