This is not an answer, but a list of references. (I can't add comments).
At least as I know, the first approach to $p$-adic fourier theory was done by Woodcock in the 70's. See "Fourier analysis for p-adic Lipschitz functions"
(J. London Math. Soc. (2) 7 (1974), 681–693). It is interesting to note that here he defines the same "integral" that Volkenborn defined also in the mid 70's. Volkenborn was inspired by some sums that Kubota and Leopoldt used to define $p$-adic zeta functions. I don't know what was the inspiration of Woodcock, but he surely was aware of $p$-adic zeta functions.
Two more advanced references are the book of Van Rooij on non-archimedean functional analysis, and Schneider and Teitelbaum $p$-adic fourier theory. The last one has many arithmetic aplications. To mention one, thay are able to prove very nice congruences between Bernoulli-Hurwitz numbers (see Katz papers for this beatiful topic).