  We find the exact radius of linearization disks at indifferent fixed points
of quadratic maps in $\mathbb{C}_p$. We also show that the radius is invariant
under power series perturbations. Localizing all periodic orbits of these
quadratic-like maps we then show that periodic points are not the only
obstruction for linearization. In so doing, we provide the first known examples
in the dynamics of polynomials over $\mathbb{C}_p$ where the boundary of the
linearization disk does not contain any periodic point.
