Any permanent progression?

[Fantoche]

Or do you restart each run naked?

[hades the gravekeeper]

naked but you can unlock upgrades (see achievements) to pick during next runs

[Auburok]

Runs are like, an hour. You progress a lot in a short amount of time.

By the end of my last run, I had firey spreadshot explosive bullets that passed through enemies and terrain that I had the option of charging up for more damage while a entorage of bots circled me.

[Slime Drips]

There's no permanent progression other than unlocking new items for appearing in the item pool and final boss kills working towards a few things. I was kinda disappointed about this since it means early on you dont really feel much sense of getting better and are just doomed to see the first area over and over, but yeah, when you are able to get a good run the whole thing takes about an hour for 100% item completion and you become a horrible mass of screen filling bullets so that's fun.

[Neocronic]

I unlock some achievements and now have chance to start with random orbs and upgrades. Last run i start with orb, who shoots 3 bullets.