  Theory of Supersymmetric Quantum Electrodynamics is extended by interactions
with external vector and tensor backgrounds, that are assumed to be generated
by some Lorentz-violating (LV) dynamics at an ultraviolet scale perhaps related
to the Planck scale. Exact supersymmetry requires that such interactions
correspond to LV operators of dimension five or higher, providing a solution to
the naturalness problem in the LV sector. We classify all dimension five and
six LV operators, analyze their properties at the quantum level and describe
observational consequences of LV in this theory. We show that LV operators do
not induce destabilizing D-terms, gauge anomaly and the Chern-Simons term for
photons. We calculate the renormalization group evolution of dimension five LV
operators and their mixing with dimension three LV operators, controlled by the
scale of the soft-breaking masses. Dimension five LV operators are constrained
by the low-energy precision measurements at 10^{-10}-10^{-5} level in units of
the inverse Planck scale, while the Planck-scale suppressed dimension six LV
operators are allowed by observational data.
