No and yes.
First, designated veteran extensions can only be signed from July to the start of the season. Second, Kyrie is not eligible for the designated veteran extension, as he was traded after his 4th year in the league. So any extension to Kyrie's contract would limit his salary to 120% of his current salary (I think).
Only way to resign Kyrie is for him to become a free agent, and the Celtics can then use his Bird rights to sign him to a new contract.
EDIT: And, yes, there is a player option for 2019-20 that would I believe kick in. Either way, Kyrie would leave too much $$$ on the table.