We explore SNe Ia as p-process sites in the framework of two-dimensional SN Ia delayed detonation and pure deflagration models. The WD precursor is assumed to have reached the Chandrasekhar mass in a binary system by mass accretion from a giant/main sequence companion. We use enhanced s-seed distributions, obtained from a sequence of thermal pulse instabilities both in the AGB phase and in the accreted material. We apply the tracer-particle method to reconstruct the nucleosynthesis by the thermal histories of Lagrangian particles, passively advected in the hydrodynamic calculations. For each particle we follow the explosive nucleosynthesis with a detailed network for all isotopes up to 209Bi. We find that SNe Ia can produce a large amount of p-nuclei, both the light p-nuclei below A=120 and the heavy-p nuclei, at quite flat average production factors, tightly related to the s-process seed distribution. For the first time, we find a stellar source able to produce both, light and heavy p-nuclei almost at the same level as 56Fe, including the very debated neutron magic 92,94Mo and 96,98Ru. We also find that there is an important contribution from p-process nucleosynthesis to the s-only nuclei 80Kr, 86Sr, to the neutron magic 90Zr, and to the neutron-rich 96Zr. Finally, we investigate the metallicity effect on p-process. Starting with different s-process seed distributions, for two metallicities Z = 0.02 and Z = 0.001, running SNe Ia models with different initial composition, we estimate that SNe Ia can contribute to, at least, 50% of the solar p-process composition.