We put this to Kat Arney.
To think about this question we really need a definition of life, whilst it is obvious that a cat is alive at a first glance it is less obvious that lichen is alive, you would only notice if you look at it for a period of years.
There are various properties that we associate with living things.
Metabolism - converting energy from one form to another. Normally with living creatures this is light or chemical energy being converted into other forms, but there is no fundamental reason life shouldn't be powered by nuclear fusion like the sun.
Homeostasis - the regulation of an internal environment, eg your body maintains its temperature by sweating, but simple creatures will regulate the chemistry in their cell. If the sun gets too hot it will expand, slowing down the nuclear reactions that power it and cooling it down so it could be said to do this.
Response to stimuli - if you metaphorically poke almost all living things they will react in some way. Some creatures will react directly to being actually poked, but even single cell creatures will react to sunlight, or food. - I guess a star will alter if you apply a large enough stimulus, but not in a way very like a living creature.
Reproduction - this is a really important one, all living things can reproduce themselves and in the process they normally have to grow. - I can't think of any way in which the sun could reproduce. Conceivably large stars that go supernova trigger the formation of other stars, but this falls down on:
Inheritance - When a living thing reproduces, the children inherit features from their parents, such as the colour of eyes, shape of foot, the ability to make an enzyme which interferes with penicillin etc. And critically this inheritance isn't quite perfect - it is liable to mutation - so the creature can evolve.
Even if you could say that a star can reproduce by going supernova the created stars do not inherit features from the star which went bang, and although a star does appear to metabolise hydrogen to form helium I don't think that you could say that a star is alive.
Saying that, the surface star is a hugely complex thing with interactions between plasmas, nuclear fusion and magnetic fields It is conceivable that there is some form of life, entirely unlike creatures on earth, and we may not recognise it even if we were looking straight at it. But at the moment this is well into the realms of science fiction.