  We prove a sharp $\sqrt{N}$ transition for the infinitesimal distribution of
a periodically banded GUE matrix. For band widths $b_N = \Omega(\sqrt{N})$, we
further prove that our model is infinitesimally free from the matrix units and
the normalized all-ones matrix. Our results allow us to extend previous work of
Shlyakhtenko on finite-rank perturbations of Wigner matrices in the
infinitesimal framework. For finite-rank perturbations of our model, we find
outliers at the classical positions from the deformed Wigner ensemble.
