  We report on a comprehensive de Haas--van Alphen (dHvA) study of the iron
pnictide LaFe$_2$P$_2$. Our extensive density-functional band-structure
calculations can well explain the measured angular-dependent dHvA frequencies.
As salient feature, we observe only one quasi-two-dimensional Fermi-surface
sheet, i.e., a hole-like Fermi-surface cylinder around $\Gamma$, essential for
$s_\pm$ pairing, is missing. In spite of considerable mass enhancements due to
many-body effects, LaFe$_2$P$_2$ shows no superconductivity. This is likely
caused by the absence of any nesting between electron and hole bands.
