Yes that is possible. There has been some discussion of it. One problem with really hot lifeforms is that the complex chains and organic reactions we have at our temperature don't work, they can't hold together. But on the other hand - we don't know too much about what happens at really hot temperatures. Most of the experiments in Earth laboratories are at room temperature or thereabouts. So we know a huge amount about the various kinds of molecules possible at a reasonably comfortable room temperature. We don't know nearly so much about what is possible at other temperatures that we only achieve rarely in our laboratory, also high pressures etc.
So for instance it's been suggested that silicon could be a basis for life at high temperatures in place of carbon. There are some problems with it. For instance, lifeforms that use oxygen and are silicon based will breath out sand instead of air - a solid rather than a gas SiO2. This seems rather hard to get working as a biochemistry. But it might well be that we just haven't been imaginative enough to understand how biochemistry would work with silicon, and don't have enough experience of high temperature work with silicon and its compounds to understand how it would work.
And this book gives a good overview of the subject, applied to the solar system, looking at many different places in the solar system where life might be possible, sometimes using unusual chemistry, for instance on Io is one exotic possibility he looks at.
But is true our focus is largely based on organic chemistry and life "as we know it" - that still gives a huge amount of variety though.
Some of the experiments suggested to search for life involve trying to cultivate it - you put a small amount of the sample into a container and then see if this has a big effect - for instance on ion concentrations - too big to explain simply by the amount introduced - if so something perhaps is reproducing in there. That's the most general kind of a test we have, the problem with it is - that it assumes you know what conditions the lifeform needs in order to reproduce, so sort of chicken and egg - if it is a rare hard to isolate lifeform - and needs exotic conditions to surprise you may not be able to cultivate it. And indeed on Earth that's not rare - indeed typically only 1% of microbes in a sample can be cultivated.
Another thing to do is to look for an environment which is out of equlibrium - if so life may be putting it out of equilibrium.
It's a multi-prong thing. But we could easily miss life, stare at it and not see it if we are looking in the wrong way. Unless it is really obvious like a whole planet of life forms, like us but based on a different biochemistry - we'd spot that. But a few exotic microbes that are based on a different kind of achemistry - or life in an unexpected medium e.g molten lava...
One objection to high temperature life is that we don't see life in molten lava. So - why not - is that not an ideal place to find it on our Earth? But it might be that it is there and for some reason we can't see it, or that it isn't on the Earth but exists in molten lava in other planets. And that is rather a special case, plentiful environment available on the Earth with no life in it as far as we know. Most extra-terrestrial environments don't occur on Earth naturally of course.