nope, no save files unless you clear a layer (level).
there IS a third party mod, not recommended by the creator (understandable, it’s in active development), that has support for three slots. it’s called “Casualties: Together” on GitHub.
it doesn’t have a Linux version, but if you wanna test it… see if it runs in Wine. it’s a unity game, so it might have some luck. only one way to find out, right?