We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^\alpha$ with $0<\alpha<1$. In fact, we can characterize the values of $\alpha$ for which such a bound holds as those exceeding $\alpha_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 3] to our purposes. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.