  We prove global existence of a derivative bi-harmonic wave equation with a
non-generic quadratic nonlinearity and small initial data in the scaling
critical space $\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times
\dot{B}^{2,1}_{\frac{d}{2}-2}(\mathbb{R}^d)$ for $ d \geq 3 $. Since the
solution persists higher regularity of the initial data, we obtain a small data
global regularity result for the biharmonic wave maps equation for a certain
class of target manifolds including the sphere.
