Maximum likelihood estimators for time-dependent mean functions within Gaussian processes are provided in the context of continuous observations. We find the widest possible class of mean functions for which the likelihood function can be written explicitly. When it is subjected to a small noise asymptotic condition leading to the vanishing of the primary Gaussian noise, we attain local asymptotic normality results, accompanied by insights into the asymptotic efficiency of these estimators. In addition, we introduce M-estimators based on discrete samples, which also leads us to the asymptotic efficiency. Furthermore, we provide quasi-information criteria for model selection analogous to Akaike Information Criteria in discretely observed cases.