****Edit based on further comment by asked*****
Man is incapble of rationalizing the concept of time without any bias. Therefore, we would be unable to actually exist without such rationalizations.
To your comment about animals. Yes, animals percieve time, even insects. The difference between nocturnal and day animals. Do they percieve it with the same complexity as humans, i doubt it. There is no reason for a horse to count days and minutes, or even hours while grazing. Huamsns like to rationalize and reason that our time must be spent in the most effective way.
Because of this, animals take the passing of the sun and moon as the equality of time. Huamsns, they do something a bit more complex; we take the aspect of time to a whole new level and ensure that our time is leveraged as much as possible.
So, No, i don't think it's possible for us to ever exist without the concept of time, mainly because of the type of reasoning and logic that we use.
****End Edit*****
It is possible for man to exist without time, yes, however, the concept is what is percieved, not what is documented.
Man, even from the earlier years has recorded time simply as the passing of the moon and sun through the sky; the passing of the moon harvest cycles, and the passing of the seaons, stars, ect.
Because of this, even in the earlier years, man has documented time as a standard as to which to live by. The additions of days, months, weeks, hours, seconds, and minutes, refines the process even further, allowing humans to maintain an accurate and precise record of time.
Can we exist without, absoutly. However, would we? No. As we 'percieve', we reason. Therefore, we percieve and reason through the aspect of time even if it didn't exist.
Thanks!