Spatial topologies are completely different from p-adic topologies in that they have the property of intersecting balls: you can have a ball at x and a ball at y with an intersection which is not equal to either ball. In p-adic balls, if two balls intersect, one is entirely inside the other. P-adic spaces are like infinite trees going down, while on the other hand, physical space is an overlapping structure, where balls overlap.

In the loop literature, where people attempt to build metrics from graphs, sometimes you do end up producing a p-adic like metric, where the intersection of ball A and ball B is either equal to ball A or ball B. This property makes it that the distance function is unphysical, it doesn't correspond to the overlap property of balls in space.

The only case in which you can get a long-distance disconnection which can be modelled as tree-like separation is if you consider the extremely large-scale structure of an eternally inflating universe, so that each little point is modelling an independent separate causal patch. This seems to be the picture you are getting at, but this picture is difficult to turn into physics, because we can only make observations within one patch, so you need to formulate the predictions in one patch for the idea to make sense in positivism.