  We prove the finiteness of formal analogues of the spherical function
(Spherical Finiteness), the ${\mathbf c}$-function (Gindikin-Karpelevich
Finiteness), and obtain a formal analogue of Harish-Chandra's limit
(Approximation Theorem) relating spherical and ${\mathbf c}$-function in the
setting of $p$-adic Kac-Moody groups. The finiteness theorems imply that the
formal analogue of the Gindikin-Karpelevich integral is well defined in local
Kac-Moody settings. These results extend the
Braverman-Garland-Kazhdan-Patnaik's affine Gindikin-Karpelevich finiteness
theorems and provide an algebraic analogue of the geometrical results of
Gaussent-Rousseau and H\'{e}bert.
