Did Mara win over Buddha?

My version of the story:

When young Gotama discovered the thorn of aversion to Samsara / thirst for Enlightenment in his heart and realized the Three Marks of Existence, his first impulse was to kill himself. He did not realize this was the very Enlightenment he sought. It took him time to recover physically and emotionally from the misery of his radical asceticism. But through understanding of his thorn he also understood the mechanism of "this/that conditionality" (idappaccayata) and the third noble truth, so he clearly understood why he had suicidal thoughts. He started reviewing his realization of the dukkha mechanism and as he did that, the retroactive realization of his Enlightenment slowly dawned on him, making him actually very glad. This was his first jhana. He then naturally fell through the rest of the four jhanas, from second to fourth, and once he was through he knew he was done-done.

Mara is personification of his crazy thoughts, the impulse of suicide being one of them. It is true that he had some of these thoughts come back post-Enlightenment, particularly some fear in connection with his human limits and some doubts about his Enlightenment. Thoughts do not belong to anyone, they arise from causes and conditions. But because he now had wisdom of Enlightenment, he had the framework for proper evaluation of thoughts, so he could no longer fall victim to them. This is why every time Buddha encounters Mara or one of the yakshas he says, you can't overturn my mind, not anymore.

So the reference to Mara is not in connection with death, it is in connection with his early suicidal thoughts. When Buddha died he died at an old age and for natural reasons, nothing to do with Mara. He was completely fulfilled as a teacher and re-discoverer of Sat Dharma. Mara did not win over him.

But yes, he certainly had hunger feelings and sexual feelings. There is a story in one sutta when he even felt a glimpse of lust towards a young woman, a daughter of one brahman who wanted him to marry her (post-Enlightenment!). But he did not like the feeling of lust in the least and dropped it immediately, to brahman's disappointment. Buddha has feelings, just like a normal person - he just does not let them control him.

In Buddhism, where is the place that says "Finally Mara wins him over by his death."

According to Buddhism, Load Buddha Win the Mara at the Boo Tree shrine.

Buddhist literature says in Maha-parinibbana Sutta: Last Days of the Buddha

63. Then the Blessed One said to the bhikkhus: "So, bhikkhus, I exhort you: All compounded things are subject to vanish. Strive with earnestness. The time of the Tathagata's Parinibbana is near. Three months hence the Tathagata will utterly pass away."

64. And having spoken these words, the Happy One, the Master, spoke again, saying:

    My years are now full ripe, the life span left is short.
    Departing, I go hence from you, relying on myself alone.
    Be earnest, then, O bhikkhus, be mindful and of virtue pure!

    With firm resolve, guard your own mind!
    Whoso untiringly pursues the Dhamma and the Discipline
    Shall go beyond the round of births and make an end of suffering.

Here Parinibbana is defer from death. After parinibbana there is no rebirth as a normal human. Others will have rebirth.

Who is Buddha?

Who remove the roots that course to sexual feelings(“Kama Thanha”) and Becoming(“Bava Thanha”) is Buddha. But He has hunger feelings until death.

** “Kama Thanha" is not just "sexual feelings". It represents all the desires generated through eyes, ears, nose, tongue, body and mind.

You misread the story. Mara did not win over the Buddha. Rather, Mara successfully influenced Ananda, to not request the Buddha to extend his life, when the Buddha dropped the hint that he could choose to extend his life, because he had the ability to do so.

Parinibbana Sutta

And the Blessed One said: "Whosoever, Ananda, has developed, practiced, employed, strengthened, maintained, scrutinized, and brought to perfection the four constituents of psychic power could, if he so desired, remain throughout a world-period or until the end of it. The Tathagata, Ananda, has done so. Therefore the Tathagata could, if he so desired, remain throughout a world-period or until the end of it."

But the Venerable Ananda was unable to grasp the plain suggestion, the significant prompting, given by the Blessed One. As though his mind was influenced by Mara, he did not beseech the Blessed One: "May the Blessed One remain, O Lord!. May the Happy One remain, O Lord, throughout the world-period, for the welfare and happiness of the multitude, out of compassion for the world, for the benefit, well being, and happiness of gods and men!"

And when for a second and a third time the Blessed One repeated his words, the Venerable Ananda remained silent.

Then the Blessed One said to the Venerable Ananda: "Go now, Ananda, and do as seems fit to you."

"Even so, O Lord." And the Venerable Ananda, rising from his seat, respectfully saluted the Blessed One, and keeping his right side towards him, took his seat under a tree some distance away.

Commentary: According to Comy., Ananda's mind had been influenced (pariyutthitacitto) by Mara's exhibiting a frightful sight which distracted his attention, preventing him from grasping the Buddha's suggestion.

No, the Blessed One renounced his will to live-on on his own volition since the bhikkhus and bhikkhunis, laymen and laywomen, have come to be true disciples... just like the paragraph says. It's not like Mara could've done anything about it, had the lord said no.

Enlightened beings feel hunger, taste, smell and other bodily sensations caused by contact as long as they have functional sense doors. But they do not get sexual feelings as it requires lust to play it's part.