We propose a Bayesian approach to detect multiple change-points in a piecewise-constant signal corrupted by a functional part corresponding to environmental or experimental disturbances. The piecewise constant part (also called segmentation part) is expressed as the product of a lower triangular matrix by a sparse vector. The functional part is a linear combination of functions from a large dictionary. A Stochastic Search Variable Selection approach is used to obtain sparse estimations of the segmentation parameters (the change-points and the means over the segments) and of the functional part. The performance of our proposed method is assessed using simulation experiments. Applications to two real datasets from geodesy and economy fields are also presented.