Bisimulation captures in a coinductive way the equivalence between processes, or trees. Several authors have defined bisimulation distances based on the bisimulation game. However, this approach becomes too local: whenever we have in one of the compared processes a large collection of branches different from those of the other, only the farthest away is taken into account to define the distance. Alternatively, we have developed a more global approach to define these distances, based on the idea of how much we need to modify one of the compared processes to obtain the other. Our original definition only covered finite processes. Instead, now we present here a coinductive approach that extends our distance to infinite but finitary trees, without needing to consider any kind of approximation of infinite trees by their finite projections.