As of Mon 2022-09-19, the Linux kernel is typically compiled with gcc -std=gnu11, which supports the GNU dialect of the 2011 edition of the ISO C standard.
Reference: https://www.kernel.org/doc/html/latest/process/programming-language.html
The kernel is written in the C programming language. More precisely,
the kernel is typically compiled with gcc under-std=gnu11: the GNU
dialect of ISO C11. clang is also supported, see docs on “Building
Linux with Clang/LLVM”.
Thanks to marc-2377 for posting a reference to that document in his answer.
Linus Torvalds has previously expressed a dislike for some features that were introduced in C99, such as mixing declarations and statements. At the time, the kernel was compiled with gcc -std=gnu89. I haven’t investigated whether his attitude has changed since then. But that’s a matter of coding style, not language standards.
I’ve deleted the bulk of my original answer since it’s no longer relevant.