The scattering problem on the Bruhat-Tits tree and its quotient spaces realizing the p-adic Riemann surfaces is studied. The spectral decomposition of the corresponding Laplace-Beltrami operator is constructed. The stationary S-matrix is obtained and the Lax-Phillips scattering theory for the problem is developed in a closed form. The Eisenstein series technique is applied to 1-loop case. The analytical structure of the scattering matrix for joint spherical functions is described.