Due to privacy and regulatory reasons, sharing data between institutions can be difficult. Because of this, real-world data are not fully exploited by machine learning (ML). An emerging method is to train ML models with federated learning (FL) which enables clients to collaboratively train ML models without sharing raw training data. We explored peer-to-peer FL by extending a prominent centralized FL algorithm called Fedavg to function in a peer-to-peer setting. We named this extended algorithm FedavgP2P. Deep neural networks at 100 simulated clients were trained to recognize digits using FedavgP2P and the MNIST data set. Scenarios with IID and non-IID client data were studied. We compared FedavgP2P to Fedavg with respect to models' convergence behaviors and communication costs. Additionally, we analyzed the connection between local client computation, the number of neighbors each client communicates with, and how that affects performance. We also attempted to improve the FedavgP2P algorithm with heuristics based on client identities and per-class F1-scores. The findings showed that by using FedavgP2P, the mean model convergence behavior was comparable to a model trained with Fedavg. However, this came with a varying degree of variation in the 100 models' convergence behaviors and much greater communications costs (at least 14.9x more communication with FedavgP2P). By increasing the amount of local computation up to a certain level, communication costs could be saved. When the number of neighbors a client communicated with increased, it led to a lower variation of the models' convergence behaviors. The FedavgP2P heuristics did not show improved performance. In conclusion, the overall findings indicate that peer-to-peer FL is a promising approach.