  We investigate spherically symmetric, static spacetimes in Eddington-inspired
Born-Infeld gravity coupled to Born-Infeld electrodynamics. The two constants,
$b^2$ and $\kappa$ which parametrise the Born-Infeld structures in the
electrodynamics (matter) and gravity sectors, characterise the features of our
analytical solutions. Black holes or naked singularities are found to arise,
depending on the values of $b^2$ and $\kappa$, as well as charge and mass.
Several such solutions are classified and understood through the analysis of
the associated metric functions for fixed $\kappa$, varying $b^2$ and
vice-versa. Further, we also compare the new metric functions with those for
the known $b^2\rightarrow \infty$ (Maxwell) and the $\kappa \rightarrow 0$
(geonic black hole) cases. Interestingly, for a particular relation between
these two parameters, $b^2=1/{4\kappa},\, \kappa >0$, we obtain a solution
resembling the well-known Reissner-Nordstr\" om line element, albeit some
modifications. Using this particular solution as the background spacetime, we
study null geodesics for Born-Infeld photons and also, gravitational lensing.
Among interesting features we note $(i)$ an increase in the radius of the
photon sphere with increasing $\kappa$ and $(ii)$ a net positive contribution
in the leading order correction term involving $\kappa$, in the weak lensing
formula for the deflection angle. We also investigate the effective potential
and light propagation for various other solutions through numerics and plots.
In summary, our work is the first attempt towards figuring out how Born-Infeld
structures in both the matter and gravity sectors can influence the nature and
character of resulting gravitational fields.
