Exiled Players/Betrayer Dead of Winter TLN
Yes.
When the players vote in favor of exiling a player, the exiled player must immediately draw 1 exiled secret objective card. This text of that card follows the following template:


If you are not the betrayer: Reveal your secret objective to all players and remove it from the game. Your new secret objective is now the following: [...]

If you are the betrayer: The 1st goal on your secret objective has changed to the following: [...]



So you must implicitly reveal that you are the betrayer by keeping your original secret objective card.