Considering the context of Functional Data Analysis (FDA), we developed and applied a new Bayesian approach for the selection of basis functions for a finite representation of functional data. The procedure makes use of a latent variable. The proposed method allows for an adaptive basis selection since it can determine the number of bases and which of them should be selected to represent the functional data. Moreover, the proposed procedure measures the uncertainty of the selection process. Our proposed methodology can also deal with curves’ random effects since the observed curves may differ not only due to experimental error but also due to random individual differences between subjects, which can be observed in a real dataset application involving daily numbers of COVID-19 cases in Brazil. A simulation study shows the proposed method’s main properties, such as its accuracy in estimating the coefficients and the strength of the procedure to find the true set of basis functions. Despite having been developed in the context of functional data analysis, the proposed model was also compared via simulation with LASSO, and Bayesian LASSO, which are methods developed for non-functional data.