We find the exact radius of linearization disks at indifferentfixed points ofquadratic maps in C_p. We also show thatthe radius is invariant under power series perturbations.Localizing all periodic orbits of these quadratic-like maps wethen show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics ofpolynomials over C_p where the boundary of the linearization disk does not contain any periodic point.