Polymer electrolytes composed of star-branched poly(ethylene oxide) PEO with addition of LiN(CF3SO2)2 salt (LiTFSI) of molar ratios EO:Li ranging from 1:1 to pure polymer were prepared by casting from solution. The electrolytes are studied by impedance spectroscopy and impedance spectroscopy performed simultaneously with optical microscope observation. Star-branched structure does not prevent the polymer without salt from crystallizing. A decrease of conductivity caused by crystallization is of similar order of magnitude as in linear PEO. Surprisingly, in electrolytes formed by addition of LiTFSI salt to the branched PEO is inhibited. In electrolytes with molar ratio EO:Li equal 10:1 and 6:1 subjected to subsequent cooling and heating runs, no traces of crystalline phase are detected by either polarizing microscope observation or differential scanning calorimetry. The ionic conductivity of studied electrolytes in amorphous state is comparable to that of electrolytes based on linear PEO. At temperatures below the melting temperature of electrolytes with linear PEO matrix, the conductivity of electrolytes based on branched PEO is much higher. As prepared, electrolyte with molar ratio of EO:Li equal 1:1 is also amorphous. Weak phase segregation effect is recorded in subsequent cooling and heating runs. This behaviour is different than observed for electrolytes based on linear PEO with similar molar ratio, which are strongly inhomogeneous and contain a large fraction of crystalline salt precipitates.