  We construct locally Lindel\"of scattered P-spaces (LLSP spaces, in short)
with prescribed widths and heights under different set-theoretic assumptions.
  We prove that there is an LLSP space of width $\omega_1$ and height
$\omega_2$ and that it is relatively consistent with ZFC that there is an LLSP
space of width $\omega_1$ and height $\omega_3$.
  Also, we prove a stepping up theorem that, for every cardinal $\lambda \geq
\omega_2$, permits us to construct from an LLSP space of width $\omega_1$ and
height $\lambda$ satisfying certain additional properties an
  LLSP space of width $\omega_1$ and height $\alpha$ for every ordinal $\alpha
< \lambda^+$. Then, we obtain as consequences of the above results the
following theorems:
  (1) For every ordinal $\alpha < \omega_3$ there is an LLSP space of width
$\omega_1$ and height $\alpha$.
  (2) It is relatively consistent with ZFC that there is an LLSP space of width
$\omega_1$ and height $\alpha$ for every ordinal $\alpha < \omega_4$.
