I have not yet played the Liszt-etude, but have succeeded with the Chopin-etude in the past. Below are my personal tips.
Before you think about stamina, make sure that your technique is efficient. With a non-ideal technique, the etude becomes significantly harder to play. I've both seen people struggle with it, who seemed really tense, and others who play it really well and seem to almost enjoy it.
In many cases, it is about minimizing unnecessary tension. And note well the word unnecessary, because obviously when we play, not all muscles are going to be relaxed. If they were, you would not be able to move your hand. However, certain muscles may become tense even though they shouldn't have to be, which does contribute to your hand getting tired. Therefore, it is important to make sure to eliminate these things before you start thinking about stamina. Think about where in your hand that you get tired, and try to figure out why.
As a specific example, consider the right hand. Something which might seem obvious is to use a rotating motion for the right hand. Doing this can significantly reduce the tension in the part of the hand below. So rather than working really hard on trying to get strong, enduring hand muscles in that part of the hand, a simple change in technique can greatly reduce the endurance needed. Maybe you already have a great technique, but if you haven't thought about it that much, it is worth looking into (surprisingly many don't).
If you have a good base technique and you are still experiencing endurance issues, I would suggest playing it through more, at the limit tempo. Others are saying that you shouldn't wear it out, which is true of course (I don't mean that you need to play it over and over again), but still, getting into the habit of "just getting through it" is a good thing. However, be careful with this. You do not want to hurt your hand, so be very cautious of pain. Whatever you do, do not let your frustration or impatience lead you into hurting your hand (let's be honest - we all experience this when struggling with technical issues). The progress you lose by seriously injuring it is way, way bigger than the progress you lose by resting for a while.
I have two suggestions for improving the endurance. The first is to play it at the limit tempo or slightly under it. If you cannot play it faster without your hand becoming tired, practice getting through it at a lower but still somewhat challenging tempo, until you can do that well. Successively increase the tempo over time.
The other suggestion is to do something similar to what you have tried - playing sections. Make the size of the sections large enough for it to be challenging. Play through it until you can play each section well and then increase the size.
But always remind yourself that you do not want to hurt your hand more than necessary. You have to have trust in yourself and the fact that you will eventually reach your goal, even though it's not happening the same day.
Lastly, you can get a significantly better result by being strategic. This essentially means knowing where to "use your power" in the etude. Sadly, there aren't many opportunities for rest in op 25 no 11, as I'm sure you are aware of. Therefore, you have to rest dynamically and "technically".
For me personally, I choose sections where I don't have to use full power and try to play these in a really relaxed and somewhat "quiet" manner. Below are some examples of how I relaxed (I used sheet music from imslp, so I'm not sure if it's the best edition. This is just to illustrate, however.)
In this part, you can of course relax a bit more, even though I wouldn't exactly play it more quietly. Subtle difference, but can actually help quite a bit. Similarly for the second time it comes, starting in C-major instead of E-major, and so on.
I would aim to not use much power at all here. It adds a nice contrast to the rest of the piece and also helps you technically.
Really use this part to fully rest your right hand. Even though there is a lot happening in the left hand, make sure that the right hand is not affected by this, so that it can "recharge its batteries".
This section is a great place to rest before the recapitulation, as I'm sure you know. I'm guessing you're already using this as a place to rest, but I do have to bring it up here, since it's such an obvious one.
Those are just some key examples of how to be strategic. Maybe you are already doing these things, but they are worth thinking about. Some people I have heard playing this seem to go in with the impression that you have to "give everything". Moreover, when they play in front of a crowd, the nervousness and pressure just makes it worse. In my opinion, it is really about distribution in the end - knowing where to use your stamina. This shouldn't have to make the interpretation worse; in fact, it enriches it.
Didn't expect this response to be this long. I got a bit carried away. But of course, the same principles hold for the Liszt etude, or any technically demanding piece.
Hope everything goes well! Best of luck!