Identification of interconnected systems is a challenging problem in which it is crucial to exploit the available knowledge about the interconnection structure. This paper addresses the identification of discrete-time dynamical models in linear fractional representation form, composed by the interconnection of a linear time-invariant block and a static nonlinearity. An iterative identification approach is adopted, which alternates the estimation of the linear and the nonlinear components. Standard identification techniques are applied to the linear part, whereas recently developed piecewise affine identification techniques are employed for modeling the static nonlinearity. The proposed method takes advantage of the interconnection structure to identify models which are more accurate and often much simpler than those obtained when applying black-box piecewise affine identification techniques to the overall system. This is demonstrated through the application of the adopted identification algorithm to the silverbox problem, a popular real-life benchmark in nonlinear system identification.