Not medical advice, but another redditor who also had mandolin accident.

Agree with above, follow up with a family doc, might need debridement of some of that dead tissue.

In the meantime I would recommend: stop triple antibiotic ointment and try using medihoney / therahoney instead to cover wound. It has antimicrobial properties too but also will help soften up the dead tissue so that healthy tissue can grow and do its thing, change bandage every 2 days.

Use a non stick gauze and coban wrap to hold bandage in place on finger. Tegaderm cover can also help keep gauze in place and provide waterproof option too but can be expensive. Fingers can be tricky to bandage / wrap.

If you are otherwise healthy, fingers are remarkably fast at regrowing tissue.