Next-generation mobile technologies are the enablers for meeting the variable specific requirements of the public safety (PS) community. In particular, due to the development of commercially deployed 5G networks, PS providers look for ways to implement related solutions upon such infrastructures, potentially in a shared use with existing commercial networks. In particular, LTE and 5G NR networks are rapidly gaining recognition as an all-inclusive critical communications platform for the delivery of both mission and business critical applications. Within this scope, we take into account the innovative architectural approach that has been proposed by the 5G ESSENCE project, addressing the paradigms of edge computing and small cell as-a-service that has been realised via a cloud-enabled small cell infrastructure leveraging multi-access technologies in 5G, where we propose a dedicated PS use case, able to offer a mission critical push-to-talk service as well as a Chat and Localisation service. Each one of these services is associated to a dedicated network slice within the scope of the corresponding fundamental 5G ESSENCE architecture and both services are provided via suitable VNFs, thus offering great flexibility to serve PS needs together. We further discuss the overall scenario deployment with the related technical enablers and the proposed functionalities, per case. The corresponding end-to-end slicing makes possible to allocate both radio and cloud resources to the involved critical actors, in an automated and elastic way.