  We initiate the study of transverse momentum-dependent (TMD) fragmentation
functions for heavy quarks, demonstrate their factorization in terms of novel
nonperturbative matrix elements in heavy-quark effective theory (HQET), and
prove new TMD sum rules that arise from heavy-quark spin symmetry. We discuss
the phenomenology of heavy-quark TMD FFs at $B$ factories and find that the
Collins effect, in contrast to claims in the literature, is not parametrically
suppressed by the heavy-quark mass. We further calculate all TMD parton
distribution functions for the production of heavy quarks from polarized gluons
within the nucleon and use our results to demonstrate the potential of the
future EIC to resolve TMD heavy-quark fragmentation in semi-inclusive DIS,
complementing the planned EIC program to use heavy quarks as probes of gluon
distributions.
