Yes, DNA can replicate without cell division.
DNA replication takes place during the synthetic stage of interphase in the cell cycle.
Cell division is a part of cell cycle in which new daughter cells are formed at the end of the divisional process of cellular contents of the cell, whereas including cell division there is one more phase of cell cycle called as interphase.
During interphase cell prepares for cell division within three main stages of it. Amongst those stages S-phase (synthetic phase) is the stage in which DNA replication occurs.
During cell division, in the late prophase those replicated DNA in the form of chromosomes become visible.
This shows that DNA replication can also takes place without cell division.