I wasn't going to post this because I thought it might already exist. But I can't find it in anyone's catalog nor a (probably not deep enough) patent search, so here it is.
Definition first, as I'm not sure if this terminology is international: Transfer tape has a backing to peel off before sticking
the tape onto something. Special dispensers exist to make application easier - they peel the backing off at the start, have a take-up roller, and a cutter (see ULINE link).
Many bandages are a kind of transfer tape. Can be tricky to peel the backing if wearing surgical gloves, or if you've just applied ointments or antiseptics and have some left on your fingertips. You also don't want to touch the sticky part or let it fold over itself. Seems like a transfer tape dispenser would be an obvious solution.
Some types of bandages have adhesives and/or medicated coating along the entire surface (e.g., silicone-based scar dressings or moisture absorbing colloids); these could be dispensed from a continuous roll. But sometimes you need gauze or other kind of padding over a wound. For those cases you might apply the pad to the wound first and roll tape over it. Or the tape itself could have pads placed at set intervals along the roll - for example, 1" pads placed a couple inches apart with adhesive just around the edges and in the space between them.
Again, I'm sure this must exist but I haven't found one. If anyone has a pointer to a close match in an existing product, I'll offer my humble thanks and close this one out.