Since it can be shown that there is a tree over 9,500 years old https://www.sciencedaily.com/releases/2008/04/080416104320.htm , and according to the dates in the Torah, mankind has been around for 5778 years, does that mean that the 6 Days of Creation were not 24-hour days?
And God said, "Let the earth sprout vegetation, seed yielding herbs and fruit trees producing fruit according to its kind in which its seed is found, on the earth," and it was so.
Thus mushrooms could not exist unless dead trees had been created for them to be groing on. Mature trees would be created with the appropriate number of tree rings. The atmosphere would have been created with the appropriate ratio of gases (including isotopes).
Carbon 14 dating assumes some initial proportion of carbon 14 to carbon 12 and that it had been decaying using that half life ever since.
As a result, it can only be use to calculate age back to the moment of creation at which point the object being dated came into existence with all of the appropriate properties.
Similarly, people argue that Adam did not have a navel because he was created fully mature (and so did not have an umbilical cord to cut) while others argue that he did because, being created fully mature, he would have shown all the evidence of how he had grown (including a navel).
As a result of this, the carbon dating process can only be accurate (to some extent) after creation and can be used only to show how Hashem wants the world to appear for ages more that. Note that 5778 is used as the time since the creation of Adam based on his life span and the following years so that it has no meaning for the time before that.
The idea is that Hashem set up the world in order to show what the laws of nature are designed to be. However, by definition, creation from nothing would have the world appear fully made with all evidence intact at the moment of creation.
A more precise wording of your question is that, as the article mentions, carbon-14 dating places a tree at over 9,500 years old, just as it places many other items much earlier than that. This is a known question about carbon-14 dating and the age of the earth.
Many people take different approaches towards this. (And this has presumably been asked elsewhere even on this site.) And yes, one popular approach (which I can agree with personally) is that the 6 Days of Creation were not 24-hour days.
Other approaches can be (a) that the Torah is false, (b) that the science is false, (c) that we don't understand something elsewhere in Biblical chronology, e.g., the long lifespans which gives us the current year as 5778.
Here is one possible alternative I devised myself, just to show you that the 6-Days as non-24 hours days is not the inexorable conclusion.
Carbon-14 dating is based on a half-life of carbon 14. Namely:
Once the organism dies, however, it ceases to absorb carbon-14, so that the amount of the radiocarbon in its tissues steadily decreases. Carbon-14 has a half-life of 5,730 ± 40 years—i.e., half the amount of the radioisotope present at any given time will undergo spontaneous disintegration during the succeeding 5,730 years. Because carbon-14 decays at this constant rate, an estimate of the date at which an organism died can be made by measuring the amount of its residual radiocarbon.
The carbon-14 method was developed by the American physicist Willard F. Libby about 1946. It has proved to be a versatile technique of dating fossils and archaeological specimens from 500 to 50,000 years old. The method is widely used by Pleistocene geologists, anthropologists, archaeologists, and investigators in related fields.
It is extrapolation based on what is currently observed as to carbon-14's decay rate. So going back only once gives us a very eerie number, 5,730 ± 40 years, which is 5770. (Reminder that this year is 5778). When the method was discovered in 1946, this was 5706. It is only when we start calculating into that second half-life that we get something that contradicts literal biblical age of the earth.
I am not saying that this is the answer, or a correct answer. Just that there are plenty of possible answers out there.