Is physical stamina important for doing mathematics? Recently, I read an interesting story about A. Weil and J.P. Serre. The general gist of the story is as follows:

During the Autumn of 1955, an international symposium on algebraic
number theory was held at Nikko [a small city around 150 km north of
Tokyo]. The Japanese mathematicians invited Weil and Serre on a trip
to Lake Chuzenji. Upon arriving, Weil and Serre stripped down, and
started swimming in the cold lake. The Japanese mathematicians
followed suit, but they quickly gave up because of the cold. After a
while, the two mathematicians came back out and started running. The
Japanese mathematicians ran after the duo, but once again, they gave
up. Eventually, Weil came back, smiled at the Japanese mathematicians,
and said, "Math is all about physical stamina."

(My translation.)
I think the last Weil quote sums up the main message of the story.
My Question. Based on people's experience, is this a true statement? If possible, I would like to hear what physical activities people pursue, and in what ways that has helped in doing mathematics.
I hope this is not too off-topic. I've never heard something like this from mathematicians I know personally, so I was curious to hear what the community thinks.
Edit. As per helpful comments, I would like to restrict the question to research mathematics (rather than mathematics in general).
 A: Mathematics is not the same as chess, but here is an anecdote from the Preface of Jan Timman's book The Art of Chess Analysis that you may find interesting.

In his Foreword [to One Hundred Selected Games], Botvinnik asks the  rhetorical question, "How do I prepare?" and he immediately answers, "That has never been any secret": fifteen to twenty days in the fresh country air, prescribes Dr. Botvinnik.
So it was that Hans Böhm and I, among others, bid farewell to our carefree lifetstyle and began a long retreat at a house in the Friesland countryside. For three months we lived like health fanatics.  Our luggage contained little more than chess literature and track suits.
The tournament began…and the first five games were lost. I remember exactly how I felt. During play my body was overflowing with so much energy that I could hardly stay seated in my chair.  After each game I still had enough energy to run several times around the Vondel Park. But why bother?
This painful start drove me to a firm decision. I threw all my Spartan habits overboard and indulged myself in everything that had been declared unhealthy. In short, I went back to my old lifestyle. And lo and behold, immediately everything went wonderfully. Thanks to a good winning streak, a total catastrophe was averted and I managed a reasonable result.
So much for that part of the wisdom I had hoped to find in Botvinnik's work. The only lesson I really learned is that you must never change your normal rhythm just because you are faced with an important tournament.  As Botvinnik says a little later in the same Foreword: "Possibly some of my suggestions will not be of much benefit to some players; each must consider them critically and apply them with caution, taking his own individual capacities and habits into account."

I should mention, for the benefit of those who do not know anything about Botvinnik, that he did indeed have a lifestyle with daily exercise built in, and he did attribute his success in part to that. So it's not just a matter of Timman misinterpreting the meaning of "fifteen to twenty days in the fresh country air."
A: Apparently this is one of those opinion-based questions that are destined to be closed soon. Still, I'll share my perspective.
IMHO it doesn't matter at all (for doing mathematics, at least) if you can outrun a cheetah or to come up as a winner in a wrestling match with an elephant. What is really required is an ability to concentrate and to think for an extended period of time without developing a headache, falling asleep, or going in circles. I doubt it burns any noticeable amount of calories because the best position I know for doing it is lying on the sofa. If that is the "stamina" we are talking about, I agree, but in all other respects, any physical shape that doesn't give you an obvious trouble and provides enough oxygenation to the brain would, probably, do.
Another possible meaningful interpretation is that perseverance is at least as important as brilliance. I'm not quite sure I believe in Edison's statement that "genius is one percent inspiration and 99 percent perspiration", but I certainly agree that trying to ride the flashes of insight alone often won't bring you very far.
As to the rest, it is in the human nature to show off a bit now and then, and, for a mathematician it is a little difficult to impress a non-mathematician (or even a mathematician in a different area) with his or her professional achievements, so I suspect many of us are in the habit of developing some marketable "side skills" and going into sports is certainly an option (though by no means the only one: somebody can choose to go into, say, arts instead and, if he ever becomes as famous as Weil, leave us pondering on MO if mathematics is all about the powers of imagination and the ability to reflect and transform all that we see and feel into sophisticated creations in the parallel world).
Just my two cents.
A: The analogy to chess reminded me of the saying that high-level chess players burn absurd numbers of calories.
Here is a place where one can find debunked the related ideas (1) that playing intense chess is comparable to intense physical activity and (2) that mental effort per se can lead to increased metabolic activity.
I have deleted the rest of the original answer which, based on assumptions (1) and (2), speculated about the implications for mathematics. Thanks to Gabe K in the comments below for challenging these speculations.
A: Having been to Japan a few times to work with my collaborators, it might well be that the hosts in your story just gave up to be polite, and would have a few laughs at the silly Westerners later in the drinking session that evening. (-:
I had a similar story. A Russian gentleman I met at a conference told me exactly the same thing Weil supposedly said. I remember him vividly because he would rise up after each talk and said something like: "Grothendieck would have done this, have you considered?". Remarkably, he asked no question after my talk, so we struck a nice conversation and went to the beach to swim (it was winter time in Korea). He was in  better shape then I was despite being about 70 (apparently people from Russia have more experience swimming in the cold).
About your actual question, obviously being fit is important in all walks of life. Keep in mind that the physical activities one enjoys depend on many factors: countries of origin, class, race, social circles, etc. Mathematicians I know tend to like walking, hiking, biking. More seem to play soccer than tennis.
Talking about Serre, here is a video of him climbing a rock a few years back, from Robin Hartshorne's FB page.
https://www.facebook.com/robin.hartshorne/videos/10200853358373274
