We saw several kayaks on our last trip. There are several spots that are deep enough that you can't touch the bottom and as previously mentioned, there will be spots you'll need to get out and carry your kayak.
Most of the time, yes you can kayak. The park is a wildlife habitat and does not permit human activity in some areas when it would impact wildlife mating, etc. If you have some skill and strength, paddling upstream is possible. There are areas you have to get out and portage.
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