  Multidimensional continued fractions (MCFs) were introduced by Jacobi and
Perron in order to obtain periodic representations for algebraic irrationals,
as it is for continued fractions and quadratic irrationals. Since continued
fractions have been also studied in the field of $p$--adic numbers $\mathbb
Q_p$, also MCFs have been recently introduced in $\mathbb Q_p$ together to a
$p$--adic Jacobi--Perron algorithm. In this paper, we address th study of two
main features of this algorithm, i.e., finiteness and periodicity. In
particular, regarding the finiteness of the $p$--adic Jacobi--Perron algorithm
our results are obtained by exploiting properties of some auxiliary integer
sequences. Moreover, it is known that a finite $p$--adic MCF represents
$\mathbb Q$--linearly dependent numbers. We see that the viceversa is not
always true and we prove that in this case infinite partial quotients of the
MCF have $p$--adic valuations equal to $-1$. Finally, we show that a periodic
MCF of dimension $m$ converges to algebraic irrationals of degree less or equal
than $m+1$ and for the case $m=2$ we are able to give some more detailed
results.
