Yes! This is possible with an on-premises deployment of Pushy Enterprise.
On iOS, Apple recently released Local Push Connectivity which finally makes it possible for your iOS users to receive push notifications in a closed network, without connectivity to APNs. This is subject to an entitlement being granted for your specific use case.
To request the entitlement to use Local Push Connectivity in your app, please submit the Entitlement Request Form on Apple's Developer Center.
On Android, Pushy can deliver notifications in a closed network using a dedicated MQTT connection.
Please contact us for more details.