Internships for PhD students (e.g. in data science or programming)
Yes. Almost every PhD student in my (US computer science) department has at least one summer internship between admission and graduation.  In fact, most PhD students go through multiple internships.

Off the top of my head, I know CS PhD students who have interned at Google (Mountain View, Manhattan, and Shanghai), Facebook, IBM, AT&amp;T, Yahoo, Motorola, Microsoft Research (Redmond, New England, and Bangalore), Disney, Zynga, Los Alamos, Sandia, Argonne, and Livermore.  Most of these places regularly advertise for interns on the web.