Psychoanalysis is generally not a fellowship, but a course of postgraduate training (that can begin during residency) that can last an undetermined number of years. Most major cities have a psychoanalytic institute (typically the center of training rather than a GME accredited hospital), and some (Boston, New York) have several.
I believe the fellowships are typically through the psychoanalytic institutes. If you just do a search on it, a bunch of sites come up. In addition to what was mentioned above, I know Baltimore, Wash DC, and UMich's psychoanaltyic institutes offer fellowships (not in the traditional sense). NYU also has a fellowship.