A basis is defined as follows:
Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components.
How does this work if we have some non-linear manifold? What is the basis of that coordinate system? This definition doesn't seem to work anymore.
You don't have a "basis" for any thing that isn't a linear space. But, as Tony S.F. said, you can define the "tangent space" of a differentiable manifold and define a basis for that.