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On creation, _pickle.Pickler caches any .persistent_id() method defined by a subclass (in the pers_func field of PicklerObject). This causes a reference cycle (pickler -> bound method of pickler -> pickler), so the pickler is held in memory until the next cycle collection. (Then, because of the pickler's memo table, any objects that this pickler has pickled are also held until the next cycle collection.)
Looking at the source code, it looks like the same thing would happen with _pickle.Unpickler and .persistent_load(), but I haven't tested it. Any fix should be applied to both classes.
I've attached a test file; when I run it with "python3 pickle_reference_cycle.py", all 3 print statements are executed. I would prefer it if "Oops, still here" was not printed. (I'm using Debian's python3.5 package, version 3.5.2-4 for amd64, but I believe the problem occurs across many versions of python3, looking at the history of _pickle.c.)
I don't see how to fix the problem with no performance impact. (Setting pers_func at the beginning of dump() and clearing it at the end would have approximately the same performance in the common case that only one object was dumped per pickler, but would be slower when dumping multiple objects.) If you decide not to fix the problem, could you at least describe the problem and a workaround in the documentation?

PR 4080 converts bound methods into unbound methods if possible.

New changeset 986375ebde0dd5ff2b7349e445a06bd28a3a8ee2 by Serhiy Storchaka in branch 'master':
bpo-28416: Break reference cycles in Pickler and Unpickler subclasses (#4080)
https://github.com/python/cpython/commit/986375ebde0dd5ff2b7349e445a06bd28a3a8ee2

New changeset c91bf742e542dceaf71042a44b5a04fb08bdda70 by Serhiy Storchaka (Miss Islington (bot)) in branch '3.6':
bpo-28416: Break reference cycles in Pickler and Unpickler subclasses (GH-4080) (#4653)
https://github.com/python/cpython/commit/c91bf742e542dceaf71042a44b5a04fb08bdda70