No, zinc is non-corrosive. Thus, so long as the zinc-steel bonded outer layer remains, paper clips will resist rusting.

Do vinyl coated paper clips rust?

DURABLE AND RUST-RESISTANT MATERIAL:

Paper clips are vinyl coated, resist corrosion, smooth surfaces prevents snags and marks or rust on papers for metal clip.

Are paper clips coated?

Colorful paperclips are properly coated, and work great.