So if I understand (please correct me if I'm wrong) Berkovich found that there was this beautiful space that you could embed the p-adics in that make analysis possible, and you're trying to show that this space is a completion of... what? The p-adics?

Something like this. To avoid technical complications (which arise from the fact that the p-adics have a discrete value group), let's assume that our field K is the (metric) completion of the algebraic closure of the p-adics.

Then, my mini-theorem says, in this case, that there is a unique (up to isomorphism) Hausdorff space X containing K as a subspace such that X has a neighborhood base of compact sets whose restrictions in K are simply the closed balls in K, and that if we take the closure of these restrictions, we get the original compact sets back.

This is saying, more or less, that X (the Berkovich space associated to K) is the universal way of forcing closed balls to be compact (note that in ultrametric fields, closed balls are both open and closed topologically). We can similarly see the real numbers as the unique space that arises when we declare that all closed intervals in the rationals should be compact. So it makes sense to think of this as some kind of completion.

Also, you said that all fields with metrics are either subfields of the complex plane or have an ultrametric. Well... are there a lot of fields out there with ultrametrics, besides the p-adics?

There are. For starters, we can take any field and put the trivial metric on it—this is a bit of a special case, since many proofs rely on the existence of elements with norm larger than one, but there's still quite a lot of theory. The complex numbers with the trivial metric have come up quite a bit in the literature.

For a more reasonable example, consider any function field F(X), where the norm of an element is e^(degree of denominator - degree of numerator). This, and related fields (power series, Laurent, and Puiseaux rings for instance), seem to come up quite a bit in tropical geometry, as sort of standard ultrametric fields that contain a given field.

Let me add one nice feature of ultrametric fields: the elements of norm less than or equal to 1 form a ring, with unique maximal ideal given by the elements of norm less than 1. The quotient is a metric-less field, called the residue field. In the case of the p-adics, this is the finite field with p elements. So not only can we make ultrametric fields out of regular fields, but we can go the other way.

Side note: as seen in the p-adics, the characteristic of the residue field can be different from the characteristic of the original field. But they can also be the same. I have seen some very interesting work that transforms fields of mixed characteristic into fields of pure characteristic and and solves old questions in number theory by doing so.