Depends on the probability distribution and the sampling procedure.
If your samples are IID then no, it is impossible. IID is an acronym for "independent and identically distributed". The "independent" part means that the probability of draw for a given sample doesn't depend on the outcome of any other sample. (E.g. rolls of a die or flips of a coin are independent. Picking a card from a deck is not independent, because it depends on what you drew from the deck earlier. Picking a card from the deck with constant replacement and reshuffling after each draw is independent.) "Identical" means the probability distribution from which a sample is drawn is the same for all samples. For example, two separate rolls of a 6-sided die is considered two identically distributed samples. If you roll a 6-sided die and an 8-sided die, these rolls are independent, but not identically distributed.
It's easy to prove that infinite IID samples from a distribution, whose probability of success is
p>0, cannot possibly have any number of finite successes. Simply take the limit of the Binomial distribution for any positive integer
k, in the limit as the number of samples approach infinity and you'll get zero. (Sorry if this paragraph was hard to read; the first sentence is really the key point, the second is only in case you want to try and prove it yourself, but can be ignored.)
You can get one positive sample out of infinite rolls if the distribution is either not independent or not identically distributed.
If it's not independent, you need only set it up as a Markov chain (which, loosely speaking, is essentially sequential probability distribution where each draw depends on earlier draws in some manner) that's set so that the probability of drawing a "success" doesn't start at zero, but can be irreversibly set to zero at some point in the sequence. (For example, this can easily be achieved by something called a "Hidden Markov Model".)
If the samples are independent, but not identical then the answer is even simpler. Only have one sample draw from a probability distribution for which a success is possible, and all other samples drawn from a distribution from which a success is not possible. Though I suspect that this scheme would violate a certain implicit assumption in your original question.