  We numerically study the spin-$1/2$ breathing kagome lattice. In this
variation of the kagome Heisenberg antiferromagnet, the spins belonging to
upward and downward facing triangles have different coupling strengths. Using
the density matrix renormalization group (DMRG) method and exact
diagonalization, we show that the kagome antiferromagnet spin liquid is
extremely robust to this anisotropy. Materials featuring this anisotropy -- and
especially the recently studied vanadium compound
$[{\mathrm{NH}}_{4}{]}_{2}[{\mathbf{C}}_{7}{\mathbf{H}}_{14}\mathbf{N}][{\mathbf{V}}_{7}{\mathbf{O}}_{6}{\mathbf{F}}_{18}]$
(DQVOF) -- may thus be very good candidates to realize the much studied kagome
spin liquid. Further, we closely examine the limit of strong breathing
anisotropy and find indications of a transition to a nematic phase.
