About the general uniformization you asked for. There is a construction by Raynaud of a general uniformization in the case of an abelian variety with semistable reduction: there exists a semiabelian variety $E$ with good reduction, extension of a good reduction abelian variety $B$ and a good reduction torus $T$:
$$ 0 \to T \to E \to B\to 0$$
(you can suppose $T$ is a split torus by extending a bit you field of definition), and analytic maps
$$ 0\to M \to E \to A \to 0$$
where $M$ is a lattice inside $E$ (the map from $E$ to $A$ is not algebraic), which you can suppose $M\cong \mathbb{Z}^r$, where $r=\dim(T)$, extending the field (as above). The case of Tate elliptic curves correspond to the case $B=0$ and $M=\mathbb{Z}$. This is what it is used by Coleman and Iovita. They are called Raynaud extension.
In the non semistable case, you can descent to a "potentially good reduction semiabelian variety" and uniformization the one above by using the semistable reduction theorem by Grothendieck (i.e., that any abelian variety acquires semistable reduction after a finite extension of you local field).
You can see:
- M. Raynaud, 1-motifs et monodromie géométrique, Exposé VII, Astérisque 223(1994) 295–319
- Bosch and Lütkebohmert, Degenerating abelian varieties, Topology 30:653–698, 1991
- Werner Lütkebohmert, Rigid Geometry of Curves and Their Jacobians, 2016, Chapter 6.
