  We derive the general conditions for a large family of shift symmetry
breaking degenerate higher order scalar-tensor (DHOST) theories to admit
stealth black hole solutions. Such black hole configurations correspond to
vacuum solutions of General Relativity and admit a scalar hair which does not
gravitate, revealing itself only at the perturbative level. We focus our
investigation on hairy Schwarzschild-(A)dS or pure Schwarzschild solutions,
dressed with a linear time-dependent scalar hair, and assuming a constant
kinetic term. We also discuss subclasses of this family which satisfy the
observational constraint $c_{\text{grav}}=c_{\text{light}}$, as well as the
recent constraint ensuring the absence of graviton decay. We provide at the end
concrete examples of DHOST lagrangians satisfying our conditions. This work
provides a first analysis of exact black hole solutions in shift symmetry
breaking DHOST theories beyond Horndeski.
