Your question is close to this one about constructing confidence limits when the binomial estimate is either zero or one. But you have multiple users, and want to estimate a small probability for each one.

Then, if you are willing to assume this small probabilities are similar (your users are exchangeable in this regard), you can get strength from using all the data at once. In some way you could estimate one common $p$, and then the individual estimates could be shrinked towards this common estimate. That could be done in a Bayesian or empirical Bayesian way. This is also called a hierarchical bayes model.

Your question is close to this one about constructing confidence limits when the binomial estimate is either zero or one. But you have multiple users, and want to estimate a small probability for each one.

Then, if you are willing to assume this small probabilities are similar (your users are exchangeable in this regard), you can get strength from using all the data at once. In some way you could estimate one common $p$, and then the individual estimates could be shrinked towards this common estimate. That could be done in a Bayesian or empirical Bayesian way. This is also called a hierarchical bayes model.

Or, since you have many users, estimate the distribution of probabilities over the users! That leads you into mixed-models.