We develop a novel Hybrid High-Order method for the incompressible Navier--Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective terms in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Denoting by $\lambda$ the $L^2$-norm of the irrotational part of the body force, the method is designed so as to mimic two key properties of the continuous problem at the discrete level, namely the invariance of the velocity with respect to λ and the non-dissipativity of the convective term. The convergence analysis shows that, when polynomials of total degree $\le k$ are used as discrete unknowns, the energy norm of the error converges as $h^{k+1}$ (with $h$ denoting, as usual, the meshsize), and the error estimate on the velocity is uniform in $\lambda$ and independent of the pressure. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard HHO formulations.