For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some exceptional ones (see e.g. wikipedia). In Paul Garrett's answer to this MSE question, after giving the non-compact forms of the infinite families of irreducible Riemannian symmetric spaces, he says

Note that in all cases the three $\mathbb{R}$-algebras $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$ play roles. In fact, a completely parallel thing happens for classical groups over $p$-adic fields and in other cases, as in A. Weil's "Algebras with involutions and classical groups", Indian (not Indiana) J. Math. 1960.

I flipped through Weil's paper but wasn't enlightened. *What I didn't find and am looking for in this question is an explicit list of p-adic homogeneous spaces $G/K$, where $K$ is presumably the fixed points of some involution, which is the analogue in whatever suitable sense of the irreducible Riemannian symmetric spaces; additionally I would like an explanation of what this 'suitable sense' is.* Any help is appreciated!

of noncompact type, the answer is probably "Euclidean building", as developed by Bruhat and Tits in the late 60's. The problem with $G/K$ is that it's just a discrete space. But the point is that it's (close to be) the 1-skeleton of a nice complex. $\endgroup$"the fact that $G/K$ is discrete isn't a problem for my purposes". I don't know what a good reference for Bruhat-Tits buildings is.) $\endgroup$