  In this paper we give an elementary proof of certain finiteness results about
affine Kac-Moody groups over a local non-archimedian field K. Our results imply
those proven earlier by Braverman-Kazhdan, Braverman-Finkelberg-Kazhdan and
Gaussent-Rousseau using either algebraic geometry or a Kac-Moody version of the
Bruhat-Tits building.
  The above finiteness results allow one to formulate an affine version of the
Gindikin-Karpelevich formula, which coincides with the one discussed by
Braverman-Finkelberg-Kazhdan in the case when K has positive characteristic. We
deduce this formula from an affine version of the Macdonald formula for the
spherical function, which will be proved in a subsequent publication.
