I think so, but I am not sure! Here is what I can tell you with confidence:

By 1592, several different people had calculated Pi well beyond seven digits. Madhava of Sangamagrama (1340 - 1425) from Kerala in India is usually attributed with having calculated Pi to 11 decimal places (3.14159265359) using the Madhava-Leibniz Series. As the name suggests, it's a infinite series discovered by him and later Leibniz (in the 1600s). Now there is some confusion here about whether Madhava himself discovered the series or one of the later people in the Keralite School of Mathematics did, as they had a habit of attributing their discoveries to Madhava, who they considered the founder of the movement. Unfortunately, I don't know enough about the controversy to tell you for sure who did it, but I do know for sure that this was published in Malayalam - the language of Kerala - by 1550 in the Yuktibhasa of Jyesthadeva, so there were at least a small group of people living in Kerala who knew pi to well beyond what you require. How many is a big question, and again we hit the limits of our small knowledge of what Keralite mathematics was. Which is quite sad, because it is a fascinating episode of history. Why exactly a small group of scholars living in one state in the south of India suddenly had an explosion of astronomical and mathematical ideas beginning in the 13th century is a very interesting question, which we still don't know a lot about. Who were they? How did they fund this? Why did they do math? How many other people knew about this? Did it spread beyond Kerala? We have a little bit about these questions, but not a lot unfortunately. The most update to intro book on this is G. G. Joseph's A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact, although it has its problems.

Of course Indians weren't using a western Calendar so the fact that there were several people with enough digits is not an answer to your question!

Ok, so if not India, where else? Jamshīd al-Kāshī head of the observatory in Samarkand under Shah Rokh calculated pi to 9 hexadecimal digits sometime before this. Wikipedia tells me 1424, and gives another source, which seems pretty plausible. Now, hexadecimal means that we're using base 16, which is familiar to computer scientists. So 3.1 = 3 + 1/16 and so on. Translating this to decimal, we get 3.14159265 35897932 so certainly enough digits. But again, the wrong calendar, as al-Kāshī would be using the Islamic calendar. And worse, it seems very likely that both al-Kāshī and Madhava never had their work make it to Europe which was using the calendar you asked for.

But, I believe we do have someone. Meet François Viète (1540–1603) Seigneur de la Bigotière, cryptographer to the King of France (and so good at it the Spanish accused him of being a witch), lawyer and mathematician. Most famous in his day for his involvement in 16th century French politics, today he's well known for math. He's the one who gave you using letters for unknowns in algebraic equations, and for our purposes, we need look at Viète's formula:

2/pi = sqrt(2)/2 + sqrt(2 + sqrt(2))/2 + sqrt(2 + sqrt(2 + sqrt(2)))/2 + ...

which he published in his book Variorum de rebus mathematicis responsorum, liber VIII, and got it to 9 decimal places. Also, he published this in 1593, however given the length of time it took to publish back then I am guessing that he probably would have known this result for some time, although I could be wrong.

So I think there was someone who could have known, but am not sure. I'm also not sure if people in the 16th century wrote dates like 3/14/1592, even today putting the month first is very American. Finally, I am aware of Viète's work because it's a rather famous result, being the first time someone published a value as the sum of some infinite series in the West (Kerala had already done this), but I don't know the state of approximations of pi beforehand. Did Europeans have up to 7 digits by then? Probably, but again I have no proof.

So my answer is I really really think so, and if Viète had just published on year earlier, I'd be sure.

Hope that helps.