It is well known that the universal covering of a complete Kahler manifold with constant bisectional curvature is $\mathbb{C}^n$, $\mathbb{B}^n$ or $\mathbb{CP}^n$. I need original paper(s) that prove this theorem.

This is theorem 7.9 in the book of KobayashiNomizu "Foundations of Differential Geometry Vol.II". There the authors attribute it to Hawley and Igusa independently. These are probably the first papers where this result was proved. Of course, as Robert Bryant points out, the correct assumption is "constant holomorphic sectional curvature". 


You asked for references to original papers. In Kobayashi and Nomizu, Vol. 2, pp. 170–171, they give a proof of this result and then write, "This has been proved independently by Hawley [1] and Igusa [1]." The two papers they cite are Hawley, N.S., Constant holomorphic curvature, Canad. J. Math 5 (1953), 53–56. (MR 14,690) and Igusa, J., On the structure of a certain class of Kähler manifolds, Amer. J. Math. 76 (1954), 669–678. (MR 16,172) Note, however, that the Math Reviews article on the Hawley paper (written by A. G. Walker) attributes the result to Bochner, but doesn't give a reference. I, myself, have never read these papers. 

