Not in an impeachment trial. This is not a court of law as we imagine such and is not bound by any rules of evidence or due process. It is a strictly political affair and can use any rules that the Senate wishes.
No it's not strictly a "political affair". Why do you think that?
The controlling authority is still the Constitution and of course the Fifth Amendment. And the enforcing authority is the Senators' affirmation that they uphold the Constitution. Could they get away with not upholding the Constitution and the Fifth Amendment during an impeachment? Yes. Does that mean they aren't supposed to uphold the Constitution and the Fifth Amendment. No it doesn't.
Because it is clearly not a judicial affair.
The hearings being run by the Democrats are a coup attempt using "Pretense-of-Law" for a very thin cover. Since when are opinions based upon 4th hand hearsay admissible as evidence of "high crimes and misdemeanors"? Since when was the right to cross-examination and the right to confront accusers eliminated? What exactly are those "high crimes and misdemeanors anyway?
The Democrats are running an open-ended fishing expedition under the principle of "Show me the man and I will show you the crime". The outcome of their "hearings" is predetermined and is some variation of "Orange Man Bad". Republicans foolish enough to attend are told to shut up and sit down. They are not allowed to do anything without permission from the Democrat Chairman.
Rule-of-Law is largely gone. We make a fatal mistake in believing that the Democrats will follow any rules at all. They certainly are not about to allow their opponents to defend themselves.
Senators commonly retire as multi-millionaires. Most of them did start out that way - and they did not make their fortunes from their salaries. It's a gravy train that all of them can ride, both Democrats and Republicans. They would like to keep that train running as long as they can.
If President Trump is impeached, the outcome will be decided by backroom deals that have nothing at all to do with legal arguments. The most important reason why there might not be a conviction, or even an impeachment, is the prospect that a number of politicians might wind up dangling from highway overpasses.
They have cut down the laws. What do they think will protect them from each other?