Setting aside the Christian retrofitting of the Enkidu and Shamhat… sorry… Adam and Eve story, it was God who told the first lie. After Adam and Eve eat the magical fruit of one tree, God has to take action to prevent them eating from a different magical tree. It is directly stated in the story that it is the fruit of the other magical tree that imparts eternal life, not refraining from eating the fruit of the first tree. (All of the quibbling about what day ‘really’ means is just a distraction.)
But the Christian mangling of the story is so tangled in knots that they can’t unravel what parts are supposed to be literal and what is metaphor. Kind of pathetic really. It’s not a very complicated story.