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Here we go. There are swimming requirements in the standards. If you can manage them you are good to go...
PADI Instructor Manual 2013:Watermanship
Before Open Water Dive 2, have student divers demonstrate that they can comfortably maintain themselves in water too deep in which to stand by completing a 10-minute swim/float without using any swim aids.
At some point before certification, have students complete a 200 metre/yard continuous surface swim or a 300 metre/ yard swim with mask, fins and snorkel.
If conditions warrant, students may wear an exposure suit as long as they are weighted for neutral buoyancy.
PADI Guide to Teaching:Watermanship
The overall objective of the 10 minute swim/float is for you, the instructor, to see that student divers have reasonable swimming ability. If conditions warrant, student divers may wear an exposure suit as long as they are weighting for neutral buoyancy.
Advise people who appear weak and uncomfortable in the water to improve through lessons or practice before engaging in dive training. Advise those with marginal water skills to practice and strengthen their skills for participation in the open water dives.
Even divers with disabilities (ie. amputee) do some sort of modified swimming under buddy supervision...