Hello, my name is***** and I have over 20 years of experience as a veterinarian. I'm sorry to hear about your girl's urinary incontinence.
It is possible that she has a urinary tract infection and the inflammation is causing her to dribble urine when she is asleep or very relaxed sitting. However I would expect her to have some symptoms when she is awake as well, such as frequent urinations, straining to urinate or licking at her vulva.
Another possibility is that she has some sort of internal organ disease (diabetes or kidney disease) that is causing her to drink more water such that she simply can't hold it for any length of time.
Other possibilities are some spinal arthritis that is affecting the nerves that control continence and causing her to dribble when she is sleeping or if she is spayed then she may have hormone related incontinence. The lack of estrogen can weaken the urethral muscles causing urine to leak when they are deeply asleep.
She needs to see her veterinarian for an examination, urinalysis with culture to look for infection and possibly some blood tests to look for abnormal organ function if her urine looks abnormal (very dilute or has protein or glucose in it).
If this is related to weak urethral muscle function due to lack of estrogen medication such as phenylpropanolamine (PPA) can help that, and if it is related to spinal arthritis then treating that with anti-inflammatories as well could help.
Please let me know if you have any further questions.