Hyperbolic graph neural networks (GNNs) have a natural advantage in modeling scale-free networks, which is a perfect fit for recommendation systems. However, existing recommendation systems based on hyperbolic GNNs often use the tangent space to achieve graph convolution on hyperbolic manifolds, which is inferior because the tangent space is only a local approximation of the manifold. In this paper, we propose a fully-hyperbolic symmetric graph neural network for collaborative filtering, named FHCF. First, to get rid of the limitation of the tangent space-based aggregation, we propose a fully-hyperbolic node aggregation operation that directly aggregates neighbors in the hyperbolic space by computing the Lorentzian centroid based on the squared Lorentzian distance. Then, to exploit the capacity of the hyperbolic space, we propose a hyperbolic space and node activity-aware push strategy to drive the node embeddings away from the origin to learn better embeddings in a wider space. Finally, to better train the model in the hyperbolic space, we further propose a symmetric hyperbolic ranking learning method, aiming at keeping negative items away from users and positive items. Extensive experimental results on three public datasets reveal that FHCF is effective and outperforms existing hyperbolic collaborative filtering models.