It is commonly assumed that surface plasmon-polariton (SPP) excitations on a metal-dielectric interface decay exponentially inside the metallic sample. Here, we show that in a wide spectral interval the SPP field decays much slower, being inversely proportional to the distance to the interface modified by an additional logarithmic factor. This dependence differs from the standard anomalous skin effect and is provisionally referred to as superanomalous. Its origin is the nonlocality and the logarithmic singularity of the dielectric permittivity in metals. This type of decay is pronounced for SPP modes of higher frequencies, but it is suppressed for light waves.