  The second-order Stark effect for a planar Dirac one-electron atom in the
ground state is analyzed within the framework of the Rayleigh-Schr\"odinger
perturbation theory, with the use of the Sturmian series expansion of the
generalized Dirac-Coulomb Green function. A closed-form analytical expression
for the static dipole polarizability of that system is found. The formula
involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit
argument. Numerical values of the polarizabilities for relativistic planar
hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in
a tabular form. A simple formula for the polarizability of a nonrelativistic
two-dimensional hydrogenic atom, reported previously by several other authors,
is recovered from our result in the nonrelativistic limit.
