Does the Vice-President retain their tiebreaking vote during an impeachment trial of the President?
No, the vice president has no defined role in the impeachment trial of the president.
It is possible that the vice president could relinquish the chair to the chief justice, but, even that is not required.
Consider also, the chief justice presides over the impeachment trial of a vice president. Allowing the vice president to have any role in their own impeachment would be a conflict of interest.
The chief justice may cast a tie-breaking vote on procedural matters. Procedure and Guidelines for Impeachment Trials in the United States Senate, Chief Justice as Presiding Officer, pp. 40-42.
The Chief Justice has voted in the case of a tie in an impeachment trial on two occasions. On March 31, 1868, a motion was made that the Senate retire for consultation. The yeas were 25 and the nays were 25, and the Chief Justice voted in the affirmative. At this point the Senate retired to its conference chamber.
During the next day’s proceedings, Senator Sumner again raised the issue of the right of the Chief Justice to vote. During the reading of the Journal, he proposed an amendment to the Journal as follows: "It appearing from the reading of the Journal of yesterday that on a question where the Senate were equally divided the Chief Justice, presiding on the trial of the President, gave a casting vote, it is hereby declared that, in the judgment of the Senate, such vote was without authority under the Constitution of the United States." This was rejected by a vote of 21 yeas, 27 nays. Thus the Senate turned down each attempt to prevent the Chief Justice from voting, and in a subsequent action concerning a motion for adjournment, the vote being yeas 22, nays 22, the Chief Justice voted in the affirmative, deciding the issue. This vote was not challenged.
At the end of the trial of President Johnson, however, another occasion arose on a motion to adjourn to a date certain when the vote was tied 27 to 27 and the Chief Justice refrained from voting.
See also How many Senators must vote to override specific judgments of the Chief Justice in an impeachment trial of the President?.