Inspired by JDługosz answer and the works of Greg Egan, I've come up with an idea that approaches the stated goals.
As far as we can tell, our universe has three dimensions of space and one dimension of time with an opposite eigenvalue to that of the spatial dimensions.
In Greg Egan's Orthogonal universe, time has the same eigenvalue, so the speed of light is dependent on its frequency, and it is possible to be living either forwards or backwards in time relative to another, as well as being able to travel infinitely fast.
To take this idea a step further, we can imagine a universe where we have two time dimensions, both with the same eigenvalues as the spatial dimensions, with a very low (average) speed of light such that the life-forms that live there can significantly change their own time-vectors.
We then have a universe in which beings aware of their own existence can have the ability to change their own 'futures'. In the event of injury, these beings would have the ability to back out of that future and head off down a different timelike path, in which the injury does not take place.
This is not to say that injury or death is unknown or impossible, just that it is an option that need not be taken, and most creatures would evolve to not take that option - those that did, wouldn't get to evolve further.
Inanimate objects could be broken and non-aware lifeforms on the order of bacteria or moulds would of course be able to be injured or die, as they have insufficient processing capability to plot a safe timelike path back from injury. However, any lifeform with sufficient brains and size - and believe me, with this kind of advantage, most lifeforms will have evolved sufficient brains and size - will be able to back out of almost any injurious or potentially lethal situation.
This is not to say that actual death or injury could not occur, just that in general, it won't (at least not without being reversed). If sufficient injury was delivered sufficiently quickly to an entire organism, the organism could be terminated so fast that it wouldn't be aware that it needed to back out of that timelike path, however, the equipment needed to do this would likely be large and obvious to anything with more than half a brain, making it unlikely to work more than a few times unless very cleverly disguised, on anything approaching sentience. It would also most likely be a technological device, and hence it's inventor could be said to have invented death.
The physiological state of being unaging is not such a stretch of the imagination - some organisms in our universe also have that trait.
This universe would have an interesting biota. There would be no predators that preyed on large organisms, as most potential prey would be smart enough to be able to back out of a predatory encounter, and since autotrophy is cheap - just emit light in a controlled way - most organisms will supply their own energy needs, and consume minerals or primitive organisms only to obtain raw materials.
Some additional features of this universe would be that electricity would be impractical, and photonics would be more likely to be used by both lifeforms and technology. Emitting light would result in an increase in an object's kinetic energy, not a decrease as in our universe. The universe itself would tend to get hotter rather than colder. There would likely be no degenerate matter or black holes.