In this paper, we study perfect codes in the Lee-Rosenbloom-Tsfasman-Jain (LRTJ) metric over the finite field Z(p). We begin by deriving some new upper bounds focusing on the number of parity check digits for linear codes correcting all error vectors of LRTJ weight up to w, 1 <= w <= 4. Furthermore, we establish sufficient conditions for the existence of perfect codes correcting all error vectors with certain weights. We also search for linear codes which attain these bounds to determine the possible parameters of perfect codes. Moreover, we derive parity check matrices corresponding linear codes correcting all error vectors of LRTJ weight 1 over Z(p), and correcting all error vectors of LRTJ weight up to 2 over Z(3) and Z(11). We also construct perfect codes for these cases. Lastly, we obtain non-existence results on w-perfect linear codes over Z(p) for 2 <= w <= 4.