Yes we have plenty.
I'm really into progressive education my daughter attend only progressive schools, she is now in college, so things have changed over the years.
In the NYC public school system they now all have to all jump through the mandatory state testing hoop (and "leave no child behind" crap). It has changed the way progressive public schools can function. But most still try hard.
The handful of progressive public schools around are very difficult to get into, high demand!
Manhattan School for Children k-8 (west 93rd street)
Central Park East (East Harlem)
East village community school
Earth School (East village)
Children's workshop school (east village)
Private school can live easily by their philosophical progressive principles.
They all give financial aid. You do not have to be broke to get aid.
The Manhattan Country School had a policy that parents pay what they can afford, parents decide on what they can pay, it is a very affordable percentage according to what they earn. Brilliant way to make private school available to many families!
In Manhattan look at these private school.
Bank Street School (created this philosophy of child centered classroom)
Little Red/Elizabeth Irwin school
Manhattan Country School
City and Country school
Rudolf Steiner School (is a Waldorf school, interesting educational pov, too teacher directed in lower grades for some parents.)