  We study a Kagome-like spin-$1/2$ Heisenberg ladder with competing
ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the
density-matrix renormalization group based calculations, we obtain the ground
state phase diagram as a function of the ratio between the FM and AFM exchange
interactions. Five different phases exist. Three of them are spin polarized
phases; an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as
FR1 and FR2 phases). The spontaneous magnetization per site is $m=1/2$, $1/3$,
and $1/6$ in the FM, FR1, and FR2 phases, respectively. This can be understood
from the fact that an effective spin-1 Heisenberg chain formed by the upper and
lower leg spins has a three-step fractional quantization of the magnetization
per site as $m=1$, $1/2$, and $0$. In particular, an anomalous "intermediate"
state $m=1/2$ of the effective spin-1 chain with the reduced Hilbert space from
SU(3) to SU(2) symmetry is highly unexpected in the context of conventional
spin-1 physics. The remaining two phases are spin-singlet phases with
translational symmetry breaking in the presence of valence bond formations. One
of them is octamer-siglet phase with a spontaneous octamerization long-range
order of the system, and the other is period-4 phase characterized by the
magnetic superstructure with a period of four structural unit cells. In these
spin-singlet phases, we find the coexistence of valence bond structure and
gapless chain. Although this may be emerged through the order-by-disorder
mechanism, there can be few examples of such a coexistence.
