Did Gautama Buddha cease to exist after death?

Any definition of Gautama Buddha after death, is not correct.

Ceased to exist, is not correct.

Still exist, is not correct.

Ceased to exist and still exist, is not correct.

Neither ceased to exist and neither still exist, is not correct.

Any definition of Gautama Buddha after death, is not correct.

Alive, is not correct.

Dead, is not correct.

Happy, is not correct.

Sad, is not correct.

Enlightened, is not correct.

Free, is not correct.

Any definition of Gautama Buddha after death, is not correct.

Why is that so?

Because after death Gautama Buddha's self does not appear.

Now you're thinking: "If self does not appear, then what appears?"

Nothing.

If nothing appears, then what remains?

He remains.

Now you're thinking: "If nothing appears, but he remains, then who is he?"

Again and again and again you're asking the same question over and over again: "After death did Gautama Buddha ceased to exist? Or does he still exist?"

Any definition we give to Gautama Buddha after death, is not correct.

If nothing appears, no definition can be given.

It's because you don't see the real nature of things that you are bewildered by my answer.

What is the real nature of things?

No self.

If there is no self, then who are you???

Meditate and gain insight about WHO YOU ARE.

If you see something, that is not you!

If you see something, that cannot be you, because the mere act of seeing something, means that is not you.

If you see nothing, that cannot be you, because the mere act of seeing nothing, means that is not you.

Seeing anything is not you. In the same way you throw a ball and see the ball flying away from you and think "I see the ball, thus the ball is not me", in the same way you see something or nothing while in meditation and think "I see something/nothing, thus something/nothing is not me".

So, who are you if there is no self to be found????

When the Buddha talked about his death, he was dealing primarily with misconceptions of death and Enlightenment. While practicing Tibetan meditation, I received initiations that are designed to allow me to dialogue with the Buddha. This suggests he is not dead. After having practiced mindfulness meditation for 50 years, I strongly suggest you use this practice to gain insights into your own issues and misconceptions. Understanding death is not important. Maybe you could read my book on mindfulness meditation.

From the Yamaka Sutta:

[Ven. Sariputta:] "And so, my friend Yamaka — when you can't pin down the Tathagata as a truth or reality even in the present life — is it proper for you to declare, 'As I understand the Teaching explained by the Blessed One, a monk with no more effluents, on the break-up of the body, is annihilated, perishes, & does not exist after death'?"

"Previously, my friend Sariputta, I did foolishly hold that evil supposition. But now, having heard your explanation of the Dhamma, I have abandoned that evil supposition, and have broken through to the Dhamma."

"Then, friend Yamaka, how would you answer if you are thus asked: A monk, a worthy one, with no more mental effluents: what is he on the break-up of the body, after death?"

"Thus asked, I would answer, 'Form is inconstant... Feeling... Perception... Fabrications... Consciousness is inconstant. That which is inconstant is stressful. That which is stressful has ceased and gone to its end."

"Very good, my friend Yamaka.

To Vacchagotta on Fire - MN 72 - https://suttacentral.net/en/mn72

Master Gotama, the monk whose mind is thus released: Where does he reappear?”

“‘Reappear,’ Vaccha, doesn’t apply.”

“In that case, Master Gotama, he does not reappear.”

“‘Does not reappear,’ Vaccha, doesn’t apply.”

“…both does & does not reappear.”

“…doesn’t apply.”

“…neither does nor does not reappear.”

“…doesn’t apply.”

“How is it, Master Gotama, when Master Gotama is asked if the monk reappears… does not reappear… both does & does not reappear… neither does nor does not reappear, he says, ‘…doesn’t apply’ in each case. At this point, Master Gotama, I am befuddled; at this point, confused. The modicum of clarity coming to me from your earlier conversation is now obscured.”

“Of course you’re befuddled, Vaccha. Of course you’re confused. Deep, Vaccha, is this phenomenon, hard to see, hard to realize, tranquil, refined, beyond the scope of conjecture, subtle, to-be-experienced by the wise. For those with other views, other practices, other satisfactions, other aims, other teachers, it is difficult to know. That being the case, I will now put some questions to you. Answer as you see fit. What do you think, Vaccha: If a fire were burning in front of you, would you know that, ‘This fire is burning in front of me’?”

“…yes…”

“And suppose someone were to ask you, Vaccha, ‘This fire burning in front of you, dependent on what is it burning?’ Thus asked, how would you reply?”

“…I would reply, ‘This fire burning in front of me is burning dependent on grass & timber as its sustenance.’”

“If the fire burning in front of you were to go out, would you know that, ‘This fire burning in front of me has gone out’?”

“…yes…”

“And suppose someone were to ask you, ‘This fire that has gone out in front of you, in which direction from here has it gone? East? West? North? Or south?’ Thus asked, how would you reply?”

“That doesn’t apply, Master Gotama. Any fire burning dependent on a sustenance of grass and timber, being unnourished—from having consumed that sustenance and not being offered any other—is classified simply as ‘out’.”

“Even so, Vaccha, any physical form by which one describing the Tathagata would describe him: That the Tathagata has abandoned, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Freed from the classification of form, Vaccha, the Tathagata is deep, boundless, hard to fathom, like the sea. ‘Reappears’ doesn’t apply. ‘Does not reappear’ doesn’t apply. ‘Both does & does not reappear’ doesn’t apply. ‘Neither reappears nor does not reappear’ doesn’t apply.

“Any feeling… Any perception… Any fabrication…

“Any consciousness by which one describing the Tathagata would describe him: That the Tathagata has abandoned, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Freed from the classification of consciousness, Vaccha, the Tathagata is deep, boundless, hard to fathom, like the sea. ‘Reappears’ doesn’t apply. ‘Does not reappear’ doesn’t apply. ‘Both does & does not reappear’ doesn’t apply. ‘Neither reappears nor does not reappear’ doesn’t apply.”

This is exactly one of the topics (i.e. whether a Tathagata exists or doesn't exist after death, or both) which the Buddha "didn't declare" -- see the Cula-Malunkyovada Sutta (MN 63):

And why are they undeclared by me? Because they are not connected with the goal, are not fundamental to the holy life.