Finnish is a real world language Philip Pullman has in a few occasions used in His Dark Materials. It is probably Serafina Pekkala's native tongue as she shows knowledge of the language and comes from the Lake Enara region, which corresponds to the northern Finland of our world, not to mention that her name as well as her dæmon's is Finnish. It is not however known if this language is known as Finnish in Lyra's world, or by some other name. In the Oxford map accompanying Lyra's Oxford there's a mention of book A Grammar of the Finno-Ugrian Languages. That term, at least, corresponds to our world usage.
It is quite doubtful if Finland exists as a nation in Lyra's world, or if there's even any unified ethnic group to be described as Finnish. Finnish nationalism arose in our world only in the 19th century, before that the speakers of Finnish were effectively divided to smaller regional groups or tribes and the land was governed by Swedish-speaking aristocracy under the crown of Sweden. In Lyra's world, Finland and other Nordic countries seem to be claimed by witch-queens and Lappish kings.
The most notable example of Finnish language in the books is Will's dæmon, to whom Serafina Pekkala gives the name Kirjava (Finnish for 'multi-coloured' or 'spotted'). A more obscure example are Nälkäinens (Finnish for 'hungry'), evil creatures of the North described to Lyra by Tony Costa. In the book Tony says simply that the name is "a northern word".