Functional integral is, by definition, an integral over a space of functions. The functions are the variables of integration. When the variables are paths, the functional integral is usually called a “path integral”.
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Primary Literature
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DeWitt-Morette, C. (2009). Functional Integration; Path Integrals. In: Greenberger, D., Hentschel, K., Weinert, F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70626-7_75
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