It's not because your playing the free version, it's because it's an early build. I'm aware content is currently limited, and more content is gonna get added. That includes floors, scenes, events, items, cards, enemies... The game is basically in a proof of concept stage.
Also, the game does have ingame saves... Were you talking about ingame *scenes* maybe? If that's the case, they're absolutely planned. Just not for this first build.