Abu Sulayman Malik ibn al-Huwayrith said,
“We came to the Prophet, may Allah bless him and grant him peace, being young men of a similar age. We spent twenty nights with him. He thought that we desired our own people and he asked us about those of our family we had left behind, and we told him. He was merciful and kind, and said, ‘Go back to your family. Instruct and command them. Pray as you have seen me praying. When it is time for the prayer, then let one of you give the adhan and let the oldest of you lead the payer.;:
And Because it is in the prayer it has to follow the “rules” of the prayer. Or do you raise hands in a prayer while you are in sujud? We are approved to do dua’ during sujud. Or do you raise hands in a prayer when you read some Verses which include a dua’ as the last 2 verses in surat al-Baqara etc.?
Abu Hurairah (May Allah be pleased with him) reported: The Messenger of Allah (ﷺ) said, “A slave becomes nearest to his Rubb (Lord) when he is in prostration. So increase supplications in prostrations.”
Therefore it seems that nobody has narrated that the Prophet (may Allah bless him and grant him peace) ever did so. And this is not allowed according to this fatwa (in Arabic).
On the other hand dua’ isn’t only a part of the prayer but also of the tawaf and sa’y and in both we are not asked to raise our hands! So raising hands for dua’ isn’t at all a standard, but it seems to be an established sunnah in cases and not in others.
Finally a fatwa in Arabic making clear where and when raising up hands for dua’ is permissible. For example it’s also not sunnah to raise hands during the dua’ of the imam at the end of the speech of jumu’a unless the imam is asking for rain.
See also this fatwa which explicitly explains when one should raise his hands for dua’ and when not.
And Allah knows best