Yes but can not go on beach or in water. Check lesh may be required
On your kayak yes but not on the beach
Per Stump Pass Beach State Park website: Pets Pets are allowed in this park, however, not on the beaches. They must be well-behaved and kept on a six-foot, hand-held leash at all times. Pets must not be left unattended, nor taken into restroom facilities or onto playground equipment. Service animals are welcome in all areas and buildings of the park, as required by law.
NO
It is a state park so I'm gonna say yes.
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