Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many application areas require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component, therefore the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally challenging due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller classification error. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.