Motivated by the rapid rise in statistical tools in Functional Data Analysis, we consider the Gaussian mechanism for achieving differential privacy with parameter estimates taking values in a, potentially infinite-dimensional, separable Banach space. Using classic results from probability theory, we show how densities over function spaces can be utilized to achieve the desired differential privacy bounds. This extends prior results of Hall et al (2013) to a much broader class of statistical estimates and summaries, including "path level" summaries, nonlinear functionals, and full function releases. By focusing on Banach spaces, we provide a deeper picture of the challenges for privacy with complex data, especially the role regularization plays in balancing utility and privacy. Using an application to penalized smoothing, we explicitly highlight this balance in the context of mean function estimation. Simulations and an application to diffusion tensor imaging are briefly presented, with extensive additions included in a supplement.