  The phenomenon of spontaneous scalarization of Reissner-Nordstr\"{o}m (RN)
black holes has recently been found in an Einstein-Maxwell-scalar (EMS) model
due to a non-minimal coupling between the scalar and Maxwell fields. Non-linear
electrodynamics, e.g., Born-Infeld (BI) electrodynamics, generalizes Maxwell's
theory in the strong field regime. Non-minimally coupling the BI field to the
scalar field, we study spontaneous scalarization of an
Einstein-Born-Infeld-scalar (EBIS) model in this paper. It shows that there are
two types of scalarized black hole solutions, i.e., scalarized RN-like and
Schwarzschild-like solutions. Although the behavior of scalarized RN-like
solutions in the EBIS model is quite similar to that of scalarize solutions in
the EMS model, we find that there exist significant differences between
scalarized Schwarzschild-like solutions in the EBIS model and scalarized
solutions in the EMS model. In particular, the domain of existence of
scalarized Schwarzschild-like solutions possesses a certain region, which is
composed of two branches. The branch of larger horizon area is a family of
disconnected scalarized solutions, which do not bifurcate from scalar-free
black holes. However, the branch of smaller horizon area may or may not
bifurcate from scalar-free black holes depending on the parameters.
Additionally, these two branches of scalarized solutions can be both
entropically disfavored over comparable scalar-free black holes in some
parameter region.
