  The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are
subgroups of $\mathbb{Z}_p^{\alpha_1} \times \mathbb{Z}_{p^2}^{\alpha_2} \times
\cdots \times \mathbb{Z}_{p^s}^{\alpha_s}$, and can be seen as linear codes
over $\mathbb{Z}_p$ when $\alpha_i=0$ for all $i \in \{2,\dots, s\}$, a
$\mathbb{Z}_{p^s}$-additive code when $\alpha_i=0$ for all $i \in \{1,\dots,
s-1\}$ , or a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code when $s=2$, or
$\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$ and $s=2$. A
$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear generalized Hadamard
(GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a
$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive code. In this
paper, we generalize some known results for
$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH codes with $p$
prime and $s\geq 2$. First, we give a recursive construction of
$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots \mathbb{Z}_{p^s}$-additive GH codes of type
$(\alpha_1,\dots,\alpha_s;t_1,\dots,t_s)$ with $t_1\geq 1,
t_2,\dots,t_{s-1}\geq 0$, and $t_s\geq1$. Then, we show for which types the
corresponding $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH
codes are nonlinear over $\mathbb{Z}_p$. We also compute the kernel and its
dimension whenever they are nonlinear.
