We analyze the N = 1 supersymmetric Wess-Zumino model dimensionally reduced to the N = 2 supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the N = (2, 2) model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the nontrivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a metric. The metric is not protected by supersymmetry and we calculate its explicit form at the fixed point. In addition we determine the exact quantum dimension of the chiral superfield and several critical exponents of interest, including the correction-to-scaling exponent omega, within the functional renormalization group approach. We compare the results obtained at different levels of truncation, exploring also a momentum-dependent wave function renormalization. Finally we briefly describe a tower of multicritical models in continuous dimensions.