  We prove new well-posedness results for dispersion-generalized
Kadomtsev--Petviashvili I equations in $\mathbb{R}^2$, which family links the
classical KP-I equation with the fifth order KP-I equation. For strong enough
dispersion, we show global well-posedness in $L^2(\mathbb{R}^2)$. To this end,
we combine resonance and transversality considerations with Strichartz
estimates and a nonlinear Loomis--Whitney inequality. Moreover, we prove that
for small dispersion, the equations cannot be solved via Picard iteration. In
this case, we use an additional frequency dependent time localization.
