  We use magnetometry, calorimetry, and high-resolution capacitive dilatometry,
as well as single-crystal neutron diffraction to explore temperature-field
phase diagram of the anisotropic honeycomb magnet BaCo$_2$(AsO$_4)_2$. Our data
reveal four distinct ordered states observed for in-plane magnetic fields. Of
particular interest is the narrow region between 0.51 and 0.55 T that separates
the up-up-down order from the fully polarized state and coincides with the
field range where signatures of the spin-liquid behavior have been reported. We
show that magnetic Bragg peaks persist in this intermediate phase, thus ruling
out its spin-liquid nature. However, the simultaneous nonmonotonic evolution of
nuclear Bragg peaks suggests the involvement of the lattice, witnessed also in
other regions of the phase diagram where large changes in the sample length are
observed upon entering the magnetically ordered states. Our data highlight the
importance of lattice effects in BaCo$_2$(AsO$_4)_2$.
