  We analyze the $\mathcal{N}=1$ supersymmetric Wess-Zumino model dimensionally
reduced to the $\mathcal{N}=2$ supersymmetric model in three Euclidean
dimensions. As in the original model in four dimensions and the
$\mathcal{N}=(2,2)$ model in two dimensions the superpotential is not
renormalized. This property puts severe constraints on the non-trivial
fixed-point solutions, which are studied in detail. We admit a field-dependent
wave function renormalization that in a geometric language relates to a
K\"ahler metric. The K\"ahler 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.
