MQTT is an application layer protocol that, thanks to its simplicity and low overhead, is widely used in the Internet of Things (IoT) devices typically found in home automation and consumer applications. The MQTT properties make it an interesting option also for Industrial IoT (IIoT) applications. However, MQTT is not specifically devised for IIoT applications requiring low-latency and the support for time-constrained transmissions. For this reason, this paper proposes an IIoT-enabled version of MQTT called a Prioritized MQTT (PrioMQTT) that is able to provide low latencies to time-critical messages. Unlike the standard MQTT, PrioMQTT adopts the UDP/IP stack, which is more suitable than TCP/IP for low-latency communications. Moreover, PrioMQTT introduces a mechanism to prioritize the time-critical messages over the non-time-critical ones. The combination of the UDP/IP stack and priority support in the PrioMQTT protocol is achieved while maintaining the compliance with the MQTT standard message format. As a result, PrioMQTT can be implemented on commercial-off-the-shelves (COTS) devices without hardware modifications. The paper describes the PrioMQTT protocol and investigates its performance through an assessment in a realistic industrial scenario and in comparison with the standard MQTT protocol.