Is there a way to have multiple players / save sets in Zelda: Breath of the Wild?

From what I've seen so far, it appears that you can only have one player at a time in Breath of The Wild. When you start the game again, the new game option prompts whether or not you want to overwrite existing games!

At least in other previous Legends of Zelda one could select which slot to save it under, but this seems to have no option for this...

Is it not possible to have multiple players going through the game at different levels of progress???

So, as was noted by @BlueBarren, both the Wii U and the Switch have the ability to have multiple profiles. Some of us have always just gone along with a singular profile with no need to have any additional profiles...

If you want to have more than one game state saved in Breath of the Wild, we can simply make a profile for each player that would like to maintain their game state and play it as that user / profile!

The first DLC that came out slightly complicated the answer to this question. You can now have 2 save files for each profile that @ylluminate mentions in their answer. One save file will be for the regular game mode, the second is for Master Mode. You can only maintain one save file in each of the two modes per profile.