In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $\Sigma_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $ n \geq 3$. For large $n$, we give an independent construction of $\Sigma_n$ and prove the existence of free boundary minimal surfaces $\tilde \Sigma_n$ in $B^3$ which have genus $1$ and $n$ boundary components. As $n$ tends to infinity, the sequence $\Sigma_n$ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of $B^3$ while the sequence $\tilde \Sigma_n$ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of $B^3-\{0\}$.