  Geodesic bulk diagrams were recently shown to be the geometric objects which
compute global conformal blocks. We show that this duality continues to hold in
$p$-adic AdS/CFT, where the bulk is replaced by the Bruhat-Tits tree, an
infinite regular graph with no cycles, and the boundary is described by
$p$-adic numbers, rather than reals. We apply the duality to evaluate the
four-point function of scalar operators of generic dimensions using tree-level
bulk diagrams. Relative to standard results from the literature, we find
intriguing similarities as well as significant simplifications. Notably, all
derivatives disappear in the conformal block decomposition of the four-point
function. On the other hand, numerical coefficients in the four-point function
as well as the structure constants take surprisingly universal forms,
applicable to both the reals and the $p$-adics when expressed in terms of local
zeta functions. Finally, we present a minimal bulk action with nearest neighbor
interactions on the Bruhat-Tits tree, which reproduces the two-, three-, and
four-point functions of a free boundary theory.
