Interesting question. It requires a definition of enlightenment, so I will provide mine and extrapolate from there.
Enlightenment is the mastery of the spiritual process. There are three operative words here. Mastery means that you can drive beneficial outcomes on your own, without help. Spiritual as opposed to mundane, mastery is obtained of transcendent abilities and qualities. Spiritual approaches work for mundane matters, but mundane approaches are often too limited to work in more abstract situations. It’s not enough to get good at one thing, it’s a whole process.
Take these three things and, in order for enlightenment to persist between lifetimes, embodiment must be added. This mastery must be made a part of your being. If your mastery is merely learned, then you will have to obtain it again once you’re reborn. It is one thing to learn something, it’s another to master it. It’s quite another indeed to have mastery so deeply imbued into being that you’d keep it even with your body, being, and person all being different.
In my model of human advancement and growth, we grow and change over the course of our lifetime. There are two main things that we build. A level of surrender, and a level of consciousness. Consciousness is the ‘aware’ side of growth, surrender is the ‘asleep’ side. All growth has conscious effort meet surrender. You do things, and things happen to you that you must learn to accept. Learning is the conscious result of the process, to be lost between lifetimes. Embodiment is the unconscious result, that which you keep with you.
To me, you don’t get another human lifetime until the next universe. After you die here, you go to the afterlife where you take what you learned here and apply it to an existence not dominated by the ugliness of physical rules. From there you move around and get to experience all that the afterlife has to offer you. Plenty of time, the universe takes billions of years to wind around to another cycle. What you do here matters, in the sense that you won’t get another do-over for awhile, but also doesn’t matter, in that the afterlife isn’t going to be all that much different for you no matter what choices you made in life.
We’ll all get to enjoy what enlightenment feels like, in this life or the next. Having it at the beginning of this life, would rob you of the joy of discovering it. Once embodied, enlightenment is simply a natural part of existence.