The Constitutional Reform and Governance Act requires peers and MPs to be tax resident and domiciled in order to remain in Parliament.
Lord Ashcroft, a Conservative deputy chairman, revealed in March he was a non-dom so did not have to pay UK taxes on most of his overseas earnings.
Five peers are now known to have quit the Lords seats to keep non-dom status.
The latest to announce the move is architect Lord Foster, who was ennobled in 1999.
The others are Conservatives  Lord Bagri, Lord McAlpine and Lord Laidlaw of Rothiemay, and cross-bencher Baroness Dunn.
The Constitutional Reform and Governance Act passed through Parliament earlier this year with cross-party support.
A three-month period during which peers could instead permanently exclude themselves from the Lords expires on Wednesday.
Labour donor Lord Paul has, like Lord Ashcroft, said he will give up his non-dom status to keep his seat.
Lord Ashcroft has given up his non-dom tax status to stay in the Lords, it has been confirmed.