One project is SeL4, which is a provably correct microkernel. I listened to a talk by one of the developers and it sounded like they were getting funding from DARPA.
https://sel4.systems/About/seL4/
I also remember reading on Wikipedia about some proprietary closed-source OS that's used by the US government to work with very highly classified information. Apparently the requirement was that the kernel and every program be formally verified, so it had very limited features. IIRC, it still maintained, but newer versions support a Linux environment for less classified work.
Unfortunately, I don't remember what it was called so I can't link to the page.
edit: found it:
https://en.wikipedia.org/wiki/XTS-400
https://en.wikipedia.org/wiki/Trusted_Computer_System_Evalua...
> Examples of A1-class systems are Honeywell's SCOMP, Aesec's GEMSOS, and Boeing's SNS Server.