There are patents, and there are trade secrets. There are no drugs that once a sample comes into possession of a competitor can't eventually be recreated by the competitor - and for something like this there is no limit to the funds they'd spend trying to duplicate the drug. In other words, the creator of the drug likely cannot create a fake scarcity or control its distribution once they start distributing it.
So you have to assume that if it ever falls into public knowledge, it will eventually become available to everyone - the rich and famous first, and ultimately it will become inexpensive enough for everyone unless there's a natural limitation or scarcity of some sort.
The government will be too late to the table unless it was discovered in their own labs.
It's probably hard to test for. We might already have a drug that extends life by 3-5 years but unless we do a long term study we might miss it. Even if a new drug comes out with similar characteristics, until hundreds of people live and die after taking it, there's no possibility of such a study. Say you develop the drug - it could take a year or more of animal testing prior to determining that it does extend life in animals, then years before it can be tested in humans. Again, this long process means that more people know about it long before it's ready.
Further, research is rarely a sudden, unexpected, and unforeseen "eureka!" moment. A lot of research takes place in deliberate steps towards a goal. Interesting results are read by other researchers and eventually someone pulls a bunch of research together, builds a hypothesis, and works towards a goal.
Lastly, human connections are very, very, very strong and often result in people choosing other people over rules and laws. In other words, once a person is aware of the drug, it's unlikely that they won't try to share it with their significant other, their children, grandchildren, and other close friends. The only thing that might overcome this is fear of getting caught, and that only if the drug is hard to make (natural scarcity) and is vigorously accounted for - which necessarily requires more people know about it in order to police each other. They would have to enter an agreement that binds their lives in the deal - immortality is great as long as you follow the rules and aren't killed by someone else also trying to protect, or worse - control - the source of immortality.
An immortality pill, therefore, wouldn't likely be controlled by the government, wouldn't likely by known only to a very small group of people, and wouldn't likely stay in a lab under lock and key without some additional forces.
So what you need to decide first is what you want to have happen, then craft the immortality pill and its creation in a way that allows your story to develop according to your desires.
For instance, introducing natural scarcity might be your biggest ally. If the process to create it is expensive and/or time consuming, such as requiring exposure in a new fusion reactor for weeks or months, or has to be created in a zero G environment for a long period of time (ie, space), or requires a particle accelerator then you may find that only a handful of pills can be created per year. It's notable that processes can also fend off competitors, particularly if the company is savvy and buys all the people or equipment available in the world that can possibly be used to create the drug. Further, as long as they don't patent the process, they can keep it a trade secret and possibly control it for far longer than scarcity obtained through other means.
If the ingredients are very rare, or do not naturally occur and must be created - such as the various radioactive elements that can only exist briefly after creation, or off-world elements (unobtanium), or materials that are government controlled (nuclear materials), etc then again, only a handful of pills might be available per year.
You'll need to decide whether the drug is something that must be taken continuously, or is a one-and-done type of situation. If it's continuous, then it can be used as a method of control, and given or denied. It also increases scarcity, but increases the number of pills available to competitors for testing and duplication.
One and done is interesting, particularly if it completely stops the aging process immediately - one would take it at their "prime" physical condition. The politics and social rules of giving them to children or babies to prevent them from ever "growing up" would be a unique topic. Perhaps it instead prevents aging beyond the prime of life, so can be given at any point prior. Perhaps it rejuvenates old cells and brings even those in their later years to a previous youthful condition. If it rejuvenates old cells, does it increase healing? Maybe if you lose a finger, or even a limb, it would eventually grow back.
Scarcity would be necessary if you wanted to make researchers and others in-the-know enter a pact of secrecy for any reason.
It's possible that the drug was discovered spontaneously, or secretly in the basement of a researcher stealing from his company. If only one person knows about it, they could take it themselves, and give it to their friends and family. They could even administer it to others secretly so no one else knew that they had become immortal.
You might also consider risks and side effects. Usually the most powerful drugs have the worst risks and side effects. Consider the Dune series - taking the spice could end up killing you, or making you more powerful. How would your world change if the drug caused death in 10-20% of those taking it, and there was no way to test for compatibility? Perhaps it causes certain physical infirmities. Acne, arthritis, reduced vision or blindness, tinnitus, mental problems such a bipolar, depression, ADD, Autism, etc.
Also, consider the impact to religious belief. Those with beliefs in the afterlife would see the drug as closing the door to their spiritual glory (for however that might be defined for them). They may see it as an offense, or blasphemy to change the laws of their deity, and may make war on those creating or using the drug.
Does the drug cause a visible effect on the person? Can you tell visually that someone has taken it and is therefore immortal? Can you tell if you have been given it - does it cause a different sensation or feeling in the body that is readily identifiable by the user?
Socially this changes what human life means - once it's widespread, and thousands of years pass, people must be getting bored. Why not enter a medically induced coma for a century or two, or based on certain events happening, such as off-world transportation or teraforming of another planet? Wars would fundamentally change. There would be an even greater pressure on the limited resources of earth, and chances are good birth control laws would be enacted (though this wouldn't be much of an issue if the drug caused infertility).
Perhaps there would be a schism between the mortals and the immortals - either due to cost of the drug, or due to religious/political beliefs. Over centuries the two would be opposing the other. Could be interesting if evolution caused the mortals to pass by the immortals in some significant way, leaving the immortals the "lessor" race.
Of course this assume the originator(s) are ethical people without any reason to mis-use the drug.
What if they instead wanted to use it for purposes of torture? "You are now immortal, and have been sentenced to 3,000 years of daily torture."
You can take this in so many directions, so a better question to ask is probably, "What impact do I want this drug to have on the world" and then design the drug to have that impact.