After toying around with Makefile.power and Makefile.system for a while, I've successfully built OpenBLAS 0.3.10 on POWER9 at Summit (ORNL) with GCC 6.4.0 (the default GCC version, at the time of writing). No luck with PGI 20.1 or IBM XL 16.1.1-5 yet, though. Both fail with the compiler cannot finding stdatomic.h.
PGI:

"../common.h", line 696: catastrophic error: cannot open source file
          "stdatomic.h"
  #include <stdatomic.h>

XL:

../common.h:696:10: fatal error: 'stdatomic.h' file not found
#include <stdatomic.h>
         ^~~~~~~~~~~~~

I wonder if this header file is part of GCC, or GlibC?

Side note:
It's interesting though that googling "PGI OpenBLAS" brings out this forum post somewhere in the first page for me, where PGI says they ship OpenBLAS with the compiler:
https://forums.developer.nvidia.com/t/auto-type-undefined-when-compiling-openblas/141268
Sure enough, in the include directory of the PGI compiler, there is a file called openblas_config.h. With PGI version 20,1, this file says

#define OPENBLAS_VERSION " OpenBLAS 0.3.7 "