  Let ${\cal C}$ be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code of length $n
> 3$. We prove that if the binary Gray image of ${\cal C}$, $C=\Phi({\cal C})$,
is a 1-perfect nonlinear code, then ${\cal C}$ cannot be a
${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic code except for one case of length
$n=15$. Moreover, we give a parity check matrix for this cyclic code. Adding an
even parity check coordinate to a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive
1-perfect code gives an extended 1-perfect code. We also prove that any such
code cannot be ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic.
