My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Keisler https://www.math.wisc.edu/~keisler/calc.html. Two of my colleagues in Belgium are similarly teaching TIC at two universities there. I am also aware of such teaching going on in France in the Strasbourg area, based on Edward Nelson's approach, though I don't have any details on that.
Which schools, colleges, or universities teach true infinitesimal calculus?
A colleague in Italy has told me about a conference a few months ago on using infinitesimals in teaching in Italian highschools. This NSA (nonstandard analysis) conference was apparently well attended (over 100 teachers showed up).
In Geneva, there are two highschools that have been teaching calculus using ultrasmall numbers for the past 10 years.
Anybody with more information about this (who to contact, what the current status of the proposal is, etc.) is hereby requested to provide such information here.
Note 1 in response to Dan's comment: usually nowadays the term infinitesimal calculus is used as a dead metaphor for "the calculus". Thus, calculus courses routinely go under the name "infinitesimal calculus" for historical reasons, whereas in point of fact no infinitesimal ever appears on the blackboard. When I refer to "true infinitesimal calculus" I mean calculus with infinitesimals (as explained above) as opposed to ordinary "infinitesimal calculus" as found in Thomas-Finney and other textbooks.
Note 2 in response to Pietro's comment: I should clarify that calculus using infinitesimals is not limited to Keisler's book. There are several books of this sort, including some that are unrelated to Robinson's theory, such as those by A. Kock and J. Bell.
Note 3: thanks to Bjørn Kjos-Hanssen for the update on the history of TIC teaching at the University of Hawaii.