Recently, Co-based honeycomb magnets have been proposed as promising candidate materials to host the Kitaev spin liquid (KSL) state. One of the front-runners is BaCo
2
(AsO
4
)
2
(BCAO), where it was suggested that the exchange processes between Co
2+
ions via the surrounding edge-sharing oxygen octahedra could give rise to bond-dependent Kitaev interactions. In this work, we present and analyze a comprehensive inelastic neutron scattering (INS) study of BCAO with fields in the honeycomb plane. Combining the constraints from the magnon excitations in the high-field polarized state and the inelastic spin structure factor measured in zero magnetic field, we examine two leading theoretical models: the Kitaev-type
JK
Γ
Γ
′
model and the XXZ
-
J
1
-
J
3
model. We show that the existing experimental data can be consistently accounted for by the XXZ
-
J
1
-
J
3
model but not by the
JK
Γ
Γ
′
model, and we discuss the implications of these results for the realization of a spin liquid phase in BCAO and more generally for the realization of the Kitaev model in cobaltates.