There is an urgent need for the development of elastic dielectric materials for flexible organic field effect transistors (OFETs). In this work, detailed analysis of the AC and DC electrical conductivity of a series of flexible poly(dimethylsiloxane) (PDMS) polymers crosslinked by metal-ligand coordination in comparison to neat PDMS was performed for the first time by means of broadband dielectric spectroscopy. The ligand was 2,2-bipyridine-4,4-dicarboxylic amide, and Ni2+, Mn2+, and Zn2+ were introduced for Cl−, Br−, and I− salts. Introduction of metal salt and creation of coordination bonds resulted in higher permittivity values increasing in an order: neat PDMS < Ni2+ < Mn2+ < Zn2+; accompanied by conductivity values of the materials increasing in an order: neat PDMS < Cl− < I− < Br−. Conductivity relaxation time plot as a function of temperature, showed Vogel-Fulcher–Tammann dependance for the Br− salts and Arrhenius type for the Cl− and I− salts. Performed study revealed that double-edged challenge can be obtained, i.e., dielectric materials with elevated value of dielectric permittivity without deterioration too much the non-conductive nature of the polymer. This opens up new perspectives for the production of flexible dielectrics suitable for gate insulators in OFETs. Among the synthesized organometallic materials, those with chlorides salts are the most promising for such applications.