In this work we present a new AFM based approach to measure the local dielectric response of polymer films at the nanoscale by means of Amplitude Modulation Electrostatic Force Microscopy (AM-EFM). The proposed experimental method is based on the measurement of the tip-sample force via the detection of the second harmonic component of the photosensor signal by means of a lock-in amplifier. This approach allows reaching unprecedented broad frequency range (2-3 × 10(4)Hz) without restrictions on the sample environment. The method was tested on different poly(vinyl acetate) (PVAc) films at several temperatures. Simple analytical models for describing the electric tip-sample interaction semi-quantitatively account for the dependence of the measured local dielectric response on samples with different thicknesses and at several tip-sample distances.