Exactly Ellen, and the arrondissement in Paris is a full administrative entity, with its city hall, civil servants, and especially its own elected mayor and municipal council. Parisian voters elect the council for their arrondissement, and the mayor of Paris is a sort of super-mayor, not directly elected by voters.
And of course a seperate town hall means a separate "bureau de l'état civil" - the municipal service in charge of recording births, deaths, marriages, among other things. So each arrondissement has its own books.