De Haan and Pereira  provided models for spatial extremes in the case of stationarity, which depend on just one parameter β > 0 measuring tail dependence, and they proposed different estimators for this parameter. This framework was supplemented in  by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold, yielding in particular asymptotic efficient estimators. The estimators investigated in these papers are based on a finite set of points t1, . . . , td, at which observations are taken. We generalize this approach in the context of functional extreme value theory (EVT). Contrary to multivariate EVT, functional EVT does not investigate finite dimensional random variables but stochastic processes. Actually it turns out that, in a proper setup, multivariate results can easily be deduced from the functional case. The more general framework of functional EVT allows estimation over some spatial parameter space, i. e., the finite set of points t1, . . . , td is replaced by t ∈ [a, b]. In particular, we derive efficient estimators of β based on those processes in a sample of iid processes in C[0, 1] which exceed a given threshold function.