Shrewsbury & Telford NHS Trust (SaTH) declared the incident at Princess Royal Hospital in Telford yesterday afternoon.
The incident does not cover the trust's other site at Royal Shrewsbury Hospital.
The trust said that the incident been declared "due to the exceptionally high demand for our urgent and emergency services and the pressure this has put on our hospital, particularly the emergency department".
A critical incident sees the trust stop some services that are considered 'non-urgent' to focus staff on areas where they are struggling to cope.
A statement said: "Despite us taking every available option to free capacity and create additional space, we need to ensure we can continue to prioritise our services and so we are pausing a very limited number of non-urgent services on this site to allow us to meet the most urgent needs of those we care for."
The trust said that urgent cancer services and "time critical procedures" were continuing.
A statement from the trust added: "We are very sorry for those impacted by this and we would like to assure you that, despite the challenges faced and some changes to non-urgent appointments, essential services remain fully open for anyone who needs them so if you require urgent medical help, please continue to come forward."
The latest critical incident follows a series declared at the trust's hospitals this year – as well as a 'system-wide' critical incident across all of the county's health services for more than two weeks in April.
Telford & Wrekin Council leader Shaun Davies said the latest critical incident was further evidence of the problems facing the county's health system.
It comes after health bosses and politicians agreed to hold a summit to look at solving the county's ambulance handover crisis – and Health Minister Edward Argar confirmed he would visit the county to see the problems first hand.
Councillor Davies said he believed more funding is needed to help NHS staff cope with demand.
He said: "From GP appointments, to ambulances to A&E to waiting lists for planned care and operations – NHS staff are working so hard yet Telford, Wrekin and Shropshire are missing out on the funding we need from the government and there is no sign of levelling up funding or extra money for social care.
"We have the minister coming to see us – we have had enough of warm words and photo opportunities we need action and we need funding now."