An interesting hypothesis from a GOP strategist:
A secret impeachment ballot might sound crazy, but it’s actually quite possible. In fact, it would take only three senators to allow for that possibility. According to current Senate procedure, McConnell will need a simple majority — 51 of the 53 Senate Republicans — to support any resolution outlining rules governing the trial. That means that if only three Republican senators were to break from the caucus, they could block any rule they didn’t like.
Those three senators, in turn, could demand a secret ballot and condition their approval of the rest of the rules on getting one. Some might say transparency in congressional deliberations and votes is inviolable, and it’s true that none of the previous Senate impeachments have been conducted via secret ballot. But the Senate’s role in an impeachment is analogous to a U.S. jury, where secret ballots are often used.
Some Republicans such as former Sen. Jeff Flake have already said that as many as 35 GOP Senators would convict Trump if they could do so in secret. Hit the link for more speculation.