How to build disk encryption system using forward permutations like Gimli? First of all, this is purely a thought experiment. The width of Gimli isn't even a power of two (384 bits), and secondary storage bus speeds aren't even worth using a high performance permutation like Gimli. So from a practical perspective, this is totally pointless. 

But it's nonetheless an interesting one. And from it, I have three concrete questions that can be considered: 


Can we build a tweakable bijective keyed permutation from just a pseudo-random forward permutation? 
Is there a practical disk encryption system using only forward permutation? 
How do we interpret the "new conventional wisdom" that "a permutation
is a better unified primitive than a block cipher"? 


Gimli paper can be here.
Disclaimer: I'm one of the authors of the said permutation.


Gimli does not aim to be used as a block cipher (in the traditional sense of it: $x \to (\sigma \circ K_{\mathit{add}})^{\mathit{nb}_{\mathit{rounds}}}$ with a block size of 384 bits or similar constructions); it is better to use it with a sponge construction such a Monkey-Duplex/Monkey-Wrap or Farfalle.
An example of such a construction would be the design of Ketje Sr.
No Idea.

  Imagine there's no blockciphers, it's easy if you try :-)
  
  
  
  Joan Daemen at FSE 2017 [abstract] [slides]


(I really advise you to have a look at the slides.)

The idea is here is to have a unique permutation (Gimli? :D) on which you can derive the rest (hash functions, CPRNG, etc.). Sure you can use a Davies–Mayer or Even–Mansour with Merkle–Damgård construction to build a hash function from a blockcipher. But using a sponge construction is more elegant and efficient: you only have one code for the permutation and the design is easier to read (thus understand and verify).