  Uniqueness results are established for time-independent finite-energy
electromagnetic fields which solve the nonlinear Maxwell--Born--Infeld
equations in boundary-free space under the condition that either the charge or
current density vanishes. In addition, it is also shown that the simpler
Maxwell--Born equations admit at most a unique stationary finite-energy
electromagnetic field solution, without the above condition. In these theories
of electromagnetism, the following physical consequences emerge: source-free
field solitons moving at speeds less than the vacuum speed of light $c$ do not
exits; any purely electrostatic (resp. magnetostatic) field is the unique
stationary electromagnetic field for the same current-density-free (resp.
charge-density-free) sources. Our results put to rest some interesting
speculations in the recent physics literature.
