To request to see a 10,000,000 year old antiproton entirely misses the point. I could just as easily ask for you to show me an electron. Or, 100 years ago, a virus. Science is only partially about viewing. It is also about collecting substantial data and understanding the story told by the entire ensemble of observations.
Antimatter, in this case antiprotons, can be created by converting energy to a combination of matter and antimatter. They have been created for decades. At Fermilab and at CERN, they have (or had) as many as 10^11 at one time. They live for arbitrarily long times...we keep a beam of antiprotons, isolated from matter, for over a day. We could keep them longer, but that is not our primary interest. Antimatter electrons (positrons) have been held in Penning traps for years.
To say that a particle lives for over 10,000,000 years does not require that you watch a particular particle for that long. You can instead look at many particles and see if ANY decay. Since decay of particles follows an exponential probability distribution (the half-life idea of radioactive decay for the non-mathematicians), by observing how many decay in a particular time period, you can determine how long a particle lives ON AVERAGE. A particular particle will live for a random amount of time...you can only determine the amount of time it is likely to live.
At any rate, by looking at the many anti protons stored at Fermilab, we can infer the average time they must live. We can state that our experiment would have seen at least one decay if the lifetime were less than 10,000,000 years. Since we have seen precisely zero antiproton decays, their lifetime must exceed that. By how much, we don't know. However our current theoretical thinking (perhaps wrong, but probably not) states that protons and antiprotons have identical lifetimes. By similar experimental techniques, we know the lifetime of protons must exceed 10^32 years. If the lifetime of the antiproton is the same, it is unsurprising that we have not seen one decay.
Finally, earlier in the thread, it was said that a particle is not stable if it needs to be held in magnetic fields. This argument contains a flaw in reasoning. While a proton is considered stable, in a laboratory consisting of antimatter, the proton would be the particle that would easily annihilate when it hit the antimatter equipment.
The real thing is that matter and antimatter are incompatible. They don't like one another and will annihilate each other and return to energy. The interesting question is "Why does the universe seem to consist primarily of matter?" This, of course, is an interesting question and not one that is properly understood. Many experiments are currently studying just this question.
The questions of the nature of creation are deep and interesting. The fact that we don't understand every aspect of it should not be converted by faulty reasoning to the statement that we don't understand anything. In fact, we understand a lot. Our studies are designed to expand that understanding. I encourage you to follow the debate, as it is all very exciting. Why should we scientists have all the fun?
Don Lincoln