Many theoretical studies were devoted in the past to ion-specific effects, trying to interpret a large body of experimental evidence, such as surface tension at air/water interfaces and force measurements between charged objects. Although several mechanisms were suggested to explain the results, such as dispersion forces and specific surface-ion interactions, we would like to suggest another source of ion-specificity, originating from the local variations of the dielectric constant due to the presence of ions in the solution. We present a mean-field model to account for the heterogeneity of the dielectric constant caused by the ions. In particular, for ions that decrease the dielectric constant we find a depletion of ions  from the vicinity of  charged surfaces. For a two-plate system, the same effect leads to an increase of the pressure in between two surfaces. Our results suggest that the effect of ions on the local dielectric constant should be taken into account when interpreting experiments that address ion-specific effects.