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We propose the functional causal Bayesian optimization method (fCBO) for finding interventions that optimize a target variable in a known causal graph. fCBO extends CBO to perform, in addition to hard interventions, functional interventions which consist in setting a variable to be a deterministic function of a set of other variables in the graph. This is achieved by modelling the unknown objective with Gaussian processes whose inputs are defined in a reproducing kernel Hilbert space, thus allowing to compute distances among vector-valued functions. In turn, this enables to sequentially select functions to explore by maximizing an expected improvement acquisition functional while keeping the typical computational tractability of standard BO settings. We show that functional interventions can attain better target effects compared to hard interventions and ensure that the found optimal policy is also optimal for sub-groups. We demonstrate the benefits of the method in a synthetic setting and in a real-world causal graph.