  We study charged scalar perturbations of charged extremal black holes in
Einstein-Born-Infeld theory. Our numerical results indicate that these black
holes all suffer from superradiant instability by the unstable quasi-bound
states, regardless how small the coupling constant is. We therefore provide a
new example that the superradiant stability of the Reissner-Nordstr\"om black
hole is a fine-tuned result, as in the case when it is embedded in the STU
supergravity model. The work is also motivated by the weak gravity conjecture
since at the linear coupling constant level, the theory belongs to a subsect of
four-derivative corrections in the effective field theory. Our results appear
to support the notion that the black holes do decay when gravity is weaker by
the correction, but the decaying halftime requires nonlinear effects and cannot
be seen at the level of linear coupling constant. The full nonlinear effects
also indicate that the black holes can decay even when gravity is stronger.
