The approach to p-adic string theory proposed by the third author [ibid. 123, No.3, 463-483 (1989; Zbl 0676.22006)] is further developed. In [loc. cit.] the p-adic analog of the interior of the open string world sheet has been proposed to be a Bruhat-Tits tree T; moreover, a simple local action on T (of gaussian type) has been constructed which reproduces the correct p-adic amplitudes. In the present paper the multiloop generalization of the above results is considered. The B-T tree is viewed as a p-adic zero genus surface. The p-adic surfaces of higher genera are obtained by factorizing this tree by some Schottky group. This gives a graph with cycles. By considering the reduced graph, i.e. the graph obtained by retaining only the cycles, one can introduce the p-adic counterpart of the Jacobian and other relevant notions. To produce the string amplitudes out of a local action one has to solve the analog of Neumann boundary problem. The problem of finding the relevant Green functions is solved within path integral approach. Having this done the authors calculate the tachyon string amplitudes.
Finally, some mathematical problems concerning the correct normalization and construction of the amplitudes for the emission of higher spin states are discussed.