The statement that Lavrov made about "Russian being banned in Ukraine" is false and untrue. It was tricky one to decipher though. This claim could stem from the law passed in 2019 that made Ukrainian the official language of Ukraine in response to Russian aggression and previous attempts to suppress Ukrainian culture and history. After the more recent 2022 invasion however there was "restrictions on the state language of the aggressor state.” "Interfax-Ukraine" a Ukrainian news agency reported on these said restrictions. Stating that "Commissioner for the Protection of the State Language, Olena Ivanovska, commenting on this petition at the signature collection stage, stated that the distribution of Russian-language book products in wartime is a challenge to cultural security, but a ban on distribution would violate the Constitution of Ukraine."
From this we can conclude 2 things, there was a restriction on Russian distribution and promotion of Russian language books in Ukraine. To go more in depth on this the restriction would also limit the import of Russian language books in Ukraine to no more the 1 unit per person. The 2nd thing coming from the latter half of the statement, a ban on distribution (of the Russian language) would be unconstitutional. Meaning that they are still upholding their belief in the constitution to allow the Russian language to not only be permitted but intertwined with their pop-culture.
As for the vocal part, the Minority Rights Group reports that it is allowed to speak Russian freely within private life. "Ukraine’s Constitution guarantees the free use of Russian and other minority languages....the Law on National Minorities (Communities). It reflects on the expanded compulsory usage of Ukrainian in the public sphere already affected by previous legislation, by introducing rules on using minority languages in ‘communication with authorities". The minority spread of language is around 34% speak Russian and 19% speak both Russian and Ukrainian. This law was not strictly for the Russian language to not be spoken as there are many different ethnic groups within Ukraine, but as a barrier against "foreign" language being spoken while "in communication with authorities".
As a whole this claim is extremely misleading, the Russian language is not banned within Ukraine, it is still widely used and pretty common within pop-culture and daily life.