  Diagrammatic many-body theory is used to calculate the scattering phase
shifts, normalized annihilation rates $Z_{\rm eff}$ and annihilation gamma
spectra for positron collisions with the hydrogen-like ions He$^+$, Li$^{2+}$,
B$^{4+}$ and F$^{8+}$. Short-range electron-positron correlations and
longer-range positron-ion correlations are accounted for by evaluating nonlocal
corrections to the annihilation vertex and the exact positron self-energy. The
numerical calculation of the many-body theory diagrams is performed using
B-spline basis sets. To elucidate the r\^ole of the positron-ion repulsion, the
annihilation rate is also estimated analytically in the Coulomb-Born
approximation. It is found that the energy dependence and magnitude of $Z_{\rm
eff}$ is governed by the Gamow factor that characterizes the suppression of the
positron wave function near the ion. For all of the H-like ions, the
correlation enhancement of the annihilation rate is found to be predominantly
due to corrections to the annihilation vertex, while the corrections to the
positron wave function play only a minor r\^ole. Results of the calculations
for $s$, $p$ and d-wave incident positrons of energies up to the positronium
formation threshold are presented. Where comparison is possible, our values are
in excellent agreement with the results obtained using other, e.g.,
variational, methods. The annihilation vertex enhancement factors obtained in
the present calculations are found to scale approximately as
$1+(1.6+0.46\ell)/Z_i$, where $Z_i$ is the net charge of the ion and $\ell$ is
the positron orbital angular momentum. Our results for positron annihilation in
H-like ions provide insights into the problem of positron annihilation with
core electrons in atoms and condensed matter systems, which have similar
binding energies.
