  Solutions of field equations in $f(R)$ gravity are found for a spherically
symmetric and static spacetime in the Born-Infeld (BI) non-linear
electrodynamics. It is found that the models supported in this configuration
must have the parametric form $f'(R)|_{r}=m+n r$, with $m,n$ constants, whose
value and sign have a strong impact on the solutions, as well as in the form
and range of $f(R)$. When $n=0$, $f(R)=m R+m_0$ and the Einstein-BI solution is
found. When $m\neq 0$ and $n\neq0$, $f(R)$ is asymptotically equivalent to GR
and the Schwarzschild and $f(R)$-Reissner-Nordstr\"om solutions are written in
some limits, likewise if $n>0$ and $r\gg1$, $f(R)$ can be found as a series
approximation and as a particular case, when $R_S=-\frac{m^2}{3n}$, explicitly
$f(R)=m R+2n\sqrt{R}+m_0$. Finally, the solutions, scalar curvature and
parametric function $f(r)$ in the non-linear ($m=0$) regime of $f(R)$ are
found, and some models for specific values of $m$ and $n$ are plotted.
