We investigate spherically symmetric, static spacetimes in Eddington-inspired Born-Infeld gravity coupled to Born-Infeld electrodynamics. The two constants, $b^2$ and $κ$ 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 $κ$, as well as charge and mass. Several such solutions are classified and understood through the analysis of the associated metric functions for fixed $κ$, 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 $κ\rightarrow 0$ (geonic black hole) cases. Interestingly, for a particular relation between these two parameters, $b^2=1/{4κ},\, κ>0$, 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 $κ$ and $(ii)$ a net positive contribution in the leading order correction term involving $κ$, 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.

19 pages, 8 figures. Revised version with changes in the section of Classification of Spacetimes and a new section for the effective potential and photon propagation. Published version