No. Although BAM mediator is synchronous as it is a mediator, data sending operation to BAM side is done asynchronously. That means if the BAM server is not reachable due to some reason (e.g., incorrect IP, BAM is not available) ESB still works properly without any problem. Only the message dumping to BAM will fail. That means mediation sequence will function properly with the server not found exception but the message logging will not happen.
During the development time of BAM mediator we considered cloning the entire message and send asynchronously to BAM side but we rejected that idea, as memory cloning will take significant time and processing which will slow down the ESB. But still data sending part (Data Bridge) works asynchronously as mentioned above.