There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there can be a theory of 'real analysis' on the p-adics. In my mind the essential characteristics of 'real analysis' over a local field $K$ of characteristic 0 necessary for this analogy to make sense are:
- There is a differential ring "$C^{\infty}(K)$" of functions $K\rightarrow K$ containing polynomials (and maybe analytic entire functions) which is closed under function composition such that the derivative is the actual limit-of-difference-quotients derivative and the kernel of the derivative is the constant functions.
- Every element of "$C^{\infty}(K)$" is the derivative of some other element (we already have uniqueness of the antiderivative up to a constant).
- There are elements of "$C^{\infty}(K)$" with bounded support (I don't want to say compact because we already know that ordinary topological notions are insufficient for some p-adic analytic ideas).
Assuming this set of properties for globally defined functions are possible you might then move on to something like a sheaf of such functions over subsets of $K$.
(I suspect such a ring exists over the p-adics because it seems like you should be able to just adjoin a single smooth 'bump function' to the ring of polynomials or analytic functions and generate the rest of the ring with the allowed operations (ring operations, composition, differentiation, and antiderivative with $f(0) = 0$ fixed) and since there are so many antiderivatives in the actual ring of smooth p-adic functions it feels like there should be an iterative axiom of choice construction that ensures you haven't generated any non-constant elements in the kernel of the derivative, but I'm not sure obviously.)
So anyways I was reminded of the construction of smooth infinitesimal analysis via toposes. I'm not very knowledgeable about them but my impression is that the construction works because you can concoct a topos in which the internal set of real valued functions is more 'well-behaved' than it normally is (i.e. every function is continuous and moreover smooth), so I'm wondering if a similar construction which would accomplish some of these goals (making the set of p-adic valued functions more 'well-behaved') is possible.