  In this paper, we formulate, for the first time, in a systematic manner,
Galilean relativistic Born-Infeld action in detail. Exploiting maps connecting
Lorentz relativistic and Galilean relativistic vectors, we construct the two
limits (electric and magnetic) of Galilean relativistic Born-Infeld action from
usual relativistic Born-Infeld theory. An action formalism is thereby derived.
From this action, equations of motion are obtained either in the potential or
field formulation. Galilean version of duality transformations involving the
electric and magnetic fields are defined. They map the electric limit relations
to the magnetic ones and vice-versa, exactly as happens for Galilean
relativistic Maxwell theory. We also explicitly show the Galilean boost and
gauge invariances of the theory in both limits.
