Theoretically, yes. I can choose to have my confidence interval be defined as (x - 0.5, x + 0.5) for an estimate x. If x represents a proportion then its value may be greater or less than 0.5 so that the interval includes values either less than 0 or greater than 1.
Is that practically useful or informative? Obviously not. This is where the concept of the inadmissibility of estimators and Uniformly Most Accurate (UMA) confidence sets comes into importance.
If a random variable exists on an interval [a, b], and an estimator, T(x), of that RV takes values outside of that interval, then the estimator
T*(x) = T, for x in [a,b]
a, for x ≤ a
b, for x ≥ b
is uniformly more accurate and has less risk (smaller MSE) and so T* should always be used instead of T.
This concept for estimators can be extended to confidence intervals. You can define any confidence interval for your random variable or parameter (there are many different ways to construct them - some better than others depending on the situation), but there is no benefit to including values or events of zero probability, so you can simply cut off those events from your interval, or use a different construction method.
