Yes definitely. We don't know enough to tell. The sun is quite young only 4.7 billion years. Compared with nearly 14 billion years old for the universe.
The younger stars are richer in the heavier elements which life uses because they are created in other older stars so they had to seed the gas clouds the stars condense from, either through supernovas or solar flares or through "planetary nebulae" where a star "shrugs off" a shell of gas, or such like.
But how much of these elements does a habitable system need? Earlier stars had less of those elements but still billlions of years before our sun, they had plenty probably for life to use.
Interstellar Medium and the Milky Way
And - there was a time when the universe was habitable everywhere at least temperature wise. 15 million years after the Big Bang, the universe had cooled down to a nice temperature between 0 and 100 C and cooled down for seven million years before it got too cold. It didn't have much by way of the heavier elements then - but even so, some really huge stars could have gone through their entire lifetime (as the largest stars burn through their fuel more quickly) with planetary nebulae and supernovae right back then - and maybe created local patches of heavier elements for life to use. Any blobs of matter in interstellar space throughout the universe would be at the right temperature for life, heated by the background radiation left over from the Big Bang.
So - that's about the earliest we know of when life could potentially have evolved at least life somewhat resembling what we have now. It wouldn't have had much time to evolve, but we don't know what is the fastest that life can evolve, don't know much about how evolution happened at all. So it's hard to say that it is impossible back then. Especially in the entire universe - was there some spot, somewhere where life began?
It's a fun idea. It might well be some time before we get decent answers to questions like this.
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