beta decay of proton-rich nuclei plays an important role in exploring isospin mixing. The beta decay of P-26 at the proton drip line is studied using double-sided silicon strip detectors operating in conjunction with high-purity germanium detectors. The T = 2 isobaric analog state (IAS) at 13 055 keV and two new high-lying states at 13 380 and 11 912 keV in Si-26 are unambiguously identified through beta-delayed two-proton emission (beta 2p). Angular correlations of two protons emitted from Si-26 excited states populated by P-26 beta decay are measured, which suggests that the two protons are emitted mainly sequentially. We report the first observation of a strongly isospin-mixed doublet that deexcites mainly via two-proton decay. The isospin mixing matrix element between the Si-26 IAS and the nearby 13 380-keV state is determined to be 130(21) keV, and this result represents the strongest mixing, highest excitation energy, and largest level spacing of a doublet ever observed in beta-decay experiments.