In the classic Euclidean setting, infinity harmonic functions are the viscosity solutions of the infinity Laplace equation \[ 0 =\Delta _\infty u=\sum\limits_{i,j = 1}^nu_{{{x_i},{x_j}}}{u_{{x_i}}}{u_{{x_j}}}. \] Note that the concept of absolutely minimizing Lipschitz extension (AMLE) makes sense in any metric space; such functions exist and are uniquely determined by their boundary values in any length space. On the other hand, the definition of infinity harmonic functions can be considered in manifolds where people have identified a way to define second-order derivatives. Moreover, the two main examples of manifolds under consideration are Riemannian spaces and Carnot-Carathéodory (also called sub-Riemannian) spaces, both of which are length spaces endowed with their natural metric. Based on these observation, the authors examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in Carnot-Carathéodory spaces. Using the weak Fubini property, they show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold.