Sure. Most of these rivalries would be either economical or cultural/influence/religion based since militarily, for much of history, it would have been much too costly to constantly wage wars far from home against an enemy that probably did not trespass on local spheres of influence, meaning a war had little to no value.

As you can imagine, the Christian nations and Islamic nations have had long standing rivalries, tensions which generally peaked during the various Crusades and subsequent Jihads.

Red Sea communities would have competed with Indian communities over the spice trade, since Europeans and Far East Asian traders would be looking to bring as much of the lucrative spices home for sale.

Rome and Carthage were fairly far flung from one another.

So distance was not a barrier to forming a long standing 'rivalry' with another faction/nation/state/kingdom/etc.

The main characteristic of such rivalries would be that they generally revolved around economics/trade or influence through culture/religion because such large distances usually ruled out territorial disputes or past military grudges.