According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$.
Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL1432.53053.
There is a nice survey by Shen and Sormani that can be found in author homepage:
Shen, Zhongmin; Sormani, Christina, The topology of open manifolds with nonnegative Ricci curvature, Commun. Math. Anal., Conference 1, 20-34 (2008). ZBL1167.53309.
And a few related and partial attacks to the conjecture:
Paeng, Seong-Hun, On the fundamental group of manifolds with almost nonnegative Ricci curvature, Proc. Am. Math. Soc. 131, No. 8, 2577-2583 (2003). ZBL1040.53042.
Xu, Senlin; Deng, Qintao, The fundamental group of open manifolds with nonnegative Ricci curvature, Acta Math. Sin., Chin. Ser. 49, No. 2, 353-356 (2006). ZBL1120.53021.
Which is stronger than
Sormani, Christina, Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups., J. Differ. Geom. 54, No. 3, 547-559 (2000). ZBL1035.53045.