Celandine grows from tiny easily dislodged bulblike roots. Every one will make a new plant and gillions of others underground. Sorry but digging them out is a herculanean task and probably not possible. What have you got against weedkiller? Paint Round up on the leaves. It will not affect any other plant like that and it is inactivated on soil contact (eventually) and it is probably the only way you will ever reduce the problem. BE glad on one thing, the celandines will have disappeared in a months time............until next year.
Personally I like them, in the right place.

It's because celandines grow from tiny 'bulbils' and if you miss a tiny piece, it grows again. Keep at it, although it's hard work, and don't let them seed, or you'll get even more.

You could use weedkiller, but you'll have to protect any precious plants from it, and it doesn't work well if it rains soon afterwards.

I'd hit 'em with glyphosphate, sprayed on to the leaves when they're actively growing, the soil is moist, but no rain is forecast for six or so hours. Wait 2 weeks, then dig - the glyphosphate seems to kill off the tiny little hairlike roots so the plants are easier to dig out - same applies to ground elder. You may need to repeat the treatment.

I have some of the different forms of them including the Bronze leafed one and best of all a white (almost) one.