  In quantum materials, the electronic interaction and the electron-phonon
coupling are, in general, two essential ingredients, the combined impact of
which may drive exotic phases. Recently, an anomalously strong
electron-electron attraction, mediated by phonons, has been unveiled in
one-dimensional copper-oxide chain Ba$_{2-x}$Sr$_x$CuO$_{3+\delta}$. Yet, it is
unclear how this strong near-neighbor attraction $V$ influences the
superconductivity pairing in the compound. Here we perform accurate many-body
calculations to study the extended Hubbard model with on-site Coulomb repulsion
$U>0$ and attraction $V<0$ that well describes the cuprate chain and likely
other similar transition-metal materials with both strong correlations and
lattice effects. We find a rich quantum phase diagram containing an intriguing
Tomonaga-Luttinger liquid phase -- besides the spin density wave and various
phase separation phases -- that can host dominant spin-triplet pairing
correlations and divergent superconductive susceptibility. Upon doping, the
spin-triplet superconducting regime can be further broadened in the parameter
space and extends to larger $U$, offering a feasible mechanism to realize
$p$-wave superconductivity in realistic cuprate chains.
