A leading model for the origin of rare proton-rich isotopes (p-isotopes) involves the transmutation of heavy elements into lighter p-isotopes via a series of (gamma, n), (gamma, p) and (gamma, alpha) reactions. Such a model, called the gamma-process, is quite successful in reproducing the abundance pattern of many rare p-isotopes if one assumes that a distribution of near-solar seeds is exposed to the thermodynamic conditions that occur naturally in the neon-oxygen layers of massive stars (M > 10-12 M.) experiencing Type II supernova explosions. However, a failure of this model is the inability to reproduce the abundance pattern of the light p-isotopes 78Kr, Mo-92, Mo-94, Ru-96, and Ru-98 due to a lack of sufficient seeds with A greater-than-or-equal-to 96. We propose that the requisite thermodynamic conditions may also occur in Type Ia or certain subclasses of Type II supernovae, where a carbon-oxygen white dwarf explodes by deflagration or detonation. For such cases, the abundance pattern of all p-nuclei, including the light p-nuclei, is well reproduced. We also note the production of several possible p-isotope cosmochronometers Mn-53 (tau-1/2: 3.7 x 10(6) yr), Nb-92 (tau-1/2: 36. x 10(6) yr), Tc-97 (tau-1/2: 2.6 x 10(6) yr) and Sm-146 (tau-1/2: 103. x 10(6) yr).