  We consider the dynamics of strongly dissipative H\'enon-like maps in the
plane, around the first bifurcation parameter $a^*$ at which the uniform
hyperbolicity is destroyed by the formation of homoclinic or heteroclinic
tangencies inside the limit set. In [Takahasi H.: Commun. Math. Phys. {\bf 312}
37-85 (2012)], it was proved that $a^*$ is a full Lebesgue density point of the
set of parameters for which Lebesgue almost every initial point diverges to
infinity under forward iteration. For these parameters, we show that all
Lyapunov exponents of all invariant ergodic Borel probability measures are
uniformly bounded away from zero, uniformly over all the parameters.
