There are multiple reasons why a child, or anyone for that matter, would have daytime sleepiness. One of the common problems is medications, such as benadryl, that can cause sleepiness. Medications can and should be reviewed with your pediatrician
regularly. If there are no likely offending medications, the investigation moves to the next step. Thinking about it simply, it makes sense that if sleep quality or quantity are poor at night, a person will have to make it up during the day.
If you are putting your daughter to bed early enough and she is falling asleep quickly at night, then the amount of sleep at night should not be the issue (children at that age should be sleeping at least 8 hours each night). The next focus then turns to sleep quality. Your question notes that your child is trying to catch her breath. An increasing problem among children (and adults) is sleep apnea, which can lead to daytime drowsiness.
Please speak with your doctor about your child's sleep issues. He or she may then refer you to a sleep specialist
or ear-nose-and throat surgeon
(aka ENT aka otolaryngologist), as this problem can often be corrected.