The structure of light proton-rich $\Lambda$-hypernuclei is addressed in the framework of the Hartree-Fock approach with effective potentials in the Skyrme form. We argue that the $^9_\Lambda \textrm{C}$ hypernucleus is bound contrary to its nuclear core $^8 \textrm{C}$. Proton-rich boron, nitrogen and oxygen hypernucleus with unstable cores are shown to remain unbound. We check also $\Lambda \Lambda$ hypernuclei with proton-rich unstable cores.