One of my Sri Lankan friends claims that Ravana spoke Sinhalese.
Is there any place in the Ramayana where the sage Valmiki mentioned about the language of Ravana and his people?
Yes, Ravana did speak Sanskrit. In fact, in this chapter of the Sundara Kanda of the Valmiki Ramayana, Hanuman hesitates before speaking to Sita in Lanka, because he's afraid that Sita will mistake him for Ravana, since Hanuman spoke the same educated Sanskrit that Ravana spoke, as Ravana was a Brahmana:
If I use Sanskrit language like a brahmin, Seetha will get frightened, thinking me as Ravana. Especially, how can a monkey speak it? ... Thereafter, this large-eyed Seetha who is full of mind, thinking me as Ravana who can assume any form at will, may shout loudly, engendered by fear.... If I eulogize Rama, who is unwearied in action and a good relation, I shall not frighten her, whose mind is directed towards that relation.