I believe that if you could also move and link files, you could actually simulate lambda calculus with a similar technique. I imagine something like this would work, where applications are described by shared prefix in same directory depth and order of application is encoded in lexicographical name order:

λx.x:

  $ tree .
  .
  └── x
      └── a -> ../x/

λsz.(s (s (s z))):

  $ tree .
  .
  └── s
      └── z
          ├── a -> ../../s/
          ├── b -> ../../s/
          ├── ca -> ../../s/
          └── cb -> ../z/