Perchance to Dream (1959)

Edward Hall visits a psychiatrist - Dr Rathmann, who was recommended by his doctor. Mr Hall tells the doctor he's very tired - since he's purposefully not slept for almost a week. Edward thinks if he sleeps, he'll die. He's been dreaming in "chapters", with 2 characters; himself, and a woman named Maya, the Cat Woman, and she's told him she'll kill him in his next nightmare.