Floor traders on the New York Stock Exchange, who once numbered in the thousands, have been pushed to the brink of extinction by the rise of electronic trading. Now, some feel they have a new nemesis: the NYSE itself.

The Big Board publicly celebrates the floor, one of the world’s last working stock-exchange trading floors and an essential part of the NYSE’s brand. Behind the scenes, conflict has broken out over a crackdown on floor traders that some say has driven smaller firms out of business.