You should be able to fit a Cauchy distribution with the Nonlinear platform, though as usual, you may have to be careful about the starting values you use.

The attached table might get you started. It has some data (in this case generated randomly from a Cauchy distribution with a median of zero), then a formula column for the negative logLikelihood that is used by Nonlinear.

The end result is shown below, where the fit is the red line: