  We consider a time-like geodesics in the background of rotating
Einstein-Born-Infeld (EBI) black hole to examine the horizon and ergosphere
structure. The effective potential that governs the particle's motion in the
spacetime and the innermost stable circular orbits (ISCO) is also studied. A
qualitative analysis is conducted to find the redshifted ultrahigh
centre-of-mass (CM) energy as a result of a two-particle collision specifically
near the horizon. The recent Event Horizon Telescope (EHT) triggered a surge of
interest in strong gravitational lensing by black holes, which provide a new
tool comparing the black hole lensing in general relativity and alternate
gravity theories. Motivated by this, we also discussed both strong and
weak-field gravitational lensing in the space-time discretely for a uniform
plasma and a singular isothermal sphere. We calculated the light deflection
coefficients $\bar{a}$ and $\bar{b}$ in the strong field limits, and their
variance with the rotational parameter $a$ for different plasma frequency as
well as in vacuum. For EBI black holes, we found that plasma's presence
increases the photon sphere radius, the deflection angle, the deflection
coefficients $\bar{a}$, $\bar{b}$, the angular positions and the angular
separation between the relativistic images. For extreme black holes, the strong
gravitational effects in the homogenous plasma are similar to those of in a
vacuum. We investigate strong gravitational lensing effects by supermassive
black holes Sgr A* and M87*. Considering rotating EBI black holes as the lens,
we find the angular position of images for Sgr A* and M87* and observe that the
deviations of the angular position from that of the analogous Kerr black hole
are not more than $2.44~\mu$as for Sgr A* and $1.83~\mu$as for M87*, which are
unlikely to get resolved by the current EHT observations.
