"living body" is correct and usable in written English.
You would use it to refer to a human or animal body which is alive. For example: "The living body requires proper nutrition to remain healthy."
"It was like operating on a living body," he said.
A studio, according to Vanderslice, is "like a living body.
A living body… We yearn for frictionless technological solutions.
Trillions of them are still alive, more than ever grew in Henrietta's living body.
When a section is removed, the building readjusts its weight distribution, like a living body.
As a living body of intellectual inquiry exploring profoundly complex questions, yes.
No technology yet conceived can supplant the transfer of experience directly from one living body to another.
As possible external signals, temperature change is one of useful stimuli due to its low invasiveness to living body system and simple site-selective application using medical devices.
As is frequently found in the living body, many vital functions are regulated by pulsed or transient release of bioactive substances at a specific site and time.
