I have had experiences with Basenjis that make me believe some do, but I would doubt it in a young pup. More likely she is starting to have issues with being left alone and the peeing is a symptom of anxiety. A video camera would tell you a lot! (is she upset when she does it or is she calm and deliberate?)
I had a similar problem when my last dog was a puppy. She was a rescue, and barely 5 weeks old when I got her, so her ability to hold her urine or bowels was not a very long length of time. As puppies get older they can go longer periods of time between eliminating, but it's important to research (google puppy housetraining) lengths of time for your puppy's age, and work out a schedule for her accordingly. My puppy needed to be brought to the puppy pad and fed every few hours around the clock, and she was initially more work than a newborn baby.
As far as the crating goes… puppies are more likely to soil a crate that is too large for them. That doesn't mean you need to run out and get a new one, but rather you can make the used area in her crate smaller. When my puppy was little, I had a soft bag carrier that was small that I placed inside the crate and closed both so she got used to the big crate, but was also just in the smaller carrier bag. You could also use boxes to reduce the size but I would worry about the puppy getting wedged in between the box and the crate, so if you use a box or wood or whatever, make sure it is secure and your puppy can NOT wiggle between it and the crate.
Also, accidents will happen with puppies once in a while. They are like human toddlers who have very little control and are learning and gaining strength every day to become more and more house trained, so patience is key.
Hope these tips help!
