In this paper, we investigate the structure sheaves of an (infinite-dimensional) affine NC-space Ancx, affine Lie-space Aliehx, and their nilpotent perturbations Anc,qx and Alieh,qx, respectively. We prove that the schemes Ancx and Aliehx are identical if and only if x is a finite set of variables, that is, when we deal with finite-dimensional noncommutative affine spaces. For each (Zariski) open subset U⊆X=Spec(C[x]), we obtain the precise descriptions of the algebras Onc(U), Onc,q(U), Olieh,q(U) and Olieh,q(U) of noncommutative regular functions on U associated with the schemes Ancx, Anc,qx, Alieh,qx and Aliehx, respectively. The obtained result for Onc(U) generalizes Kapranov's formula in the finite-dimensional case. Our approach to the matter is based on a noncommutative holomorphic functional calculus in Fréchet algebras.