Spatially varying coefficient (SVC) models are a type of regression models for spatial data where covariate effects vary over space. If there are several covariates, a natural question is which covariates have a spatially varying effect and which not. We present a new variable selection approach for Gaussian process-based SVC models. It relies on a penalized maximum likelihood estimation and allows joint variable selection both with respect to fixed effects and Gaussian process random effects. We validate our approach in a simulation study as well as a real world data set. In the simulation study, the penalized maximum likelihood estimation correctly identifies zero fixed and random effects, while the penalty-induced bias of non-zero estimates is negligible. In the real data application, our proposed penalized maximum likelihood estimation yields sparser SVC models and achieves a smaller information criterion than classical maximum likelihood estimation. In a cross-validation study applied on the real data, we show that our proposed penalized maximum likelihood estimation consistently yields the sparsest SVC models while achieving similar predictive performance compared to other SVC modeling methodologies.