Can't save in Star Fox 64?

Is there a way to save your progress or do you have to start from the begining each time?

you can't, that's the game's style

I'm hopeful that they add a save system to the 3DS remake with it being a portable system, but yeah, on N64 the game is short enough that you can play through it in one session. Always better to let you save than not these days, but at the time it probably just wasn't considered necessary.

No save function in Star Fox 64, but you do have a pause function.

Yeah, you can't save in Starfox 64.