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This is to confirm understanding about ssh, whether ssh fails if server RAM/Swap memory is exhausted.
Is server RAM/Swap space has any role in ssh connection? If yes, could you please explain in detail.

Recently we faced such issue where server RAM/Swap was exhausted and client was unable to connect with server using ssh.

Most programs nowadays use dynamic memory allocation, which means they allocate memory on demand. SSH server is not an exception. But I don't think that the problem is the SSH server itself since
handling the connection is probably not making the server request another memory page.

When a user log in through SSH, a new process will be started to hold the actual shell. At least it will require a single memory page, which won't be available if the system is saturated.
Even if you could sucessfully open a shell, by reusing one for example, you wouldnt be able to do anything but using the internal shell commands, since the system couldnt start a new process.

Intel® Xeon 2.53GHz [...]