Did Former United Nations Secretary-General Ban Ki-Moon say that Ukraine does not exist? No, that's not true: there is no such statement in open sources.
The claim appeared in 2014 in several Russian sources, including Livejournal. Nine years later it resurfaced as a video (archived here) where it was posted on TikTok by a user named @recidivist121 on Apr. 30, 2023, under the title "Ukraine has no borders." It stated:
Ukraine has no borders. They are neither declared nor registered with the UN. In accordance with the CIS treaty, Ukraine is an administrative district of the USSR. Russia can only make a statement and declare Ukraine its territory since Ukraine was part of the USSR and the Russian Federation is the legal successor of the USSR. There is no sovereign state of Ukraine and never has been.
This is what the post looked like on TikTok at the time of writing:
Ban made comments that were reflected on the United Nations website from Apr. 7, 2014, but did not mention Ukrainian borders. Instead, Ban Ki-Moon expressed concern about the situation in the east of Ukraine. There is no statement or speech where he says Ukraine does not exist.
Furthermore, the United Nations official website says, "The United Nations is neither a State nor a Government, and therefore does not possess any authority to recognize either a State or a Government. As an organization of independent States, it may admit a new State to its membership or accept the credentials of the representatives of a new Government." Ukraine is listed as a member state of the UN.
In 1991, Ukraine changed its status from the Ukrainian Soviet Socialist Republic to Ukraine. There were no requirements to register the borders of the country.
(Source: TikTok screenshot taken on Sun May 7 15:35:11 2023 UTC)