We develop the functional It\^o/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter \(H> \frac{1}{2}\). Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second type is Wick-It\^o integral. Then we establish the functional It\^o formulas for fractional Brownian motion, which extend the functional It\^o formulas in Dupire (2009) and Cont-Fourni\'e (2013) to the case of non-semimartingale. Finally, as an application, we deal with a class of fractional backward stochastic differential equations (BSDEs). A relation between fractional BSDEs and path-dependent partial differential equations (PDEs) is established.