The electrical conductivity of highly concentrated solutions of tetra-n-butylammonium picrate (TBAP) in 1-dodecanol (dielectric constant epsilon =4.6) and 1,4-butanediol (epsilon =25.9), and in mixtures of both alcohols, is measured in an extended temperature range approximate to 10(-5)<< approximate to 10(-1), where tau=(T-T-c)/T-c is the reduced temperature with T-c, the critical temperature. The electrical conductivity Lambda (T) obeys the Vogel-Fulcher-Tammann (VFT) law for the temperatures far from the critical one. In the temperature range tau < 10(-2) a systematic deviation of the electrical conductivity from the regular VFT behavior is observed. This deviation is attributed to a critical anomaly. At the critical point the amplitude of the critical anomaly is finite with a value which varies between approximate to0.4 and approximate to2.7% of Lambda (T-c), depending on the solvent. The (1-alpha) critical exponent describes well the conductivity anomaly, alpha being the exponent of the specific heat anomaly at constant pressure. The value of the Walden product (Lambda (eqv)eta), with Lambda (eqv), the equivalent conductivity and eta, the shear viscosity, allows the degree of dissociation alpha (diss) of TBAP to be determined at the critical point. alpha (diss) becomes larger for increasing values of epsilon: for TBAP in 1-dodecanol alpha (diss)approximate to0.25 and in 1,4-butanediol alpha (diss)approximate to0.73. When the degree of dissociation of the salt is accounted for the Debye screening length is found almost independent on epsilon.