In this paper, we consider a semiparametric single-index panel data model with cross-sectional depen- dence and stationarity. Meanwhile, we allow fixed effects to be correlated with the regressors to capture unobservable heterogeneity. Under a general spatial error dependence structure, we then establish some consistent closed-form estimates for both the unknown parameters and the link function for the case where both cross-sectional dimension (N) and temporal dimension (T) go to infinity. Rates of conver- gence and asymptotic normality are established for the proposed estimates. Our experience suggests that the proposed estimation method is simple and thus attractive for finite-sample studies and empirical im- plementations. Moreover, both the finite-sample performance and the empirical applications show that the proposed estimation method works well when the cross-sectional dependence exists in the data set.