[usgroup]

x ~ Binomial(N,p) and you wish to estimate p.

Here are a whole collection methods for how to estimate p and calculate a confidence interval for it: https://en.wikipedia.org/wiki/Binomial_distribution#Confiden...

One of the methods is Bayesian; the rest are not.

Not mentioned in the list, but you can also use likelihood ratio intervals calculated from a likelihood profile: another Frequentist method.

None of the methods -- including the Bayesian, requires an informative prior.