It would be helpful if you could specify a bit further what you mean by a "chase" narrative. What elements are desirable? What should be excluded? From a single modern example it's impossible to predict exactly what you mean. There's no use citing ancient examples if you're going to say "No, that doesn't count, because of this thing I didn't mention previously." That happens with monotonous regularity with this form of question. I wouldn't even want to try guessing at which other modern movies you think qualify as parallels, let alone ancient myths!

Having said that, it's hard to think of any mythological narratives at all where an entire story is structured around a chase of any kind. Isolated motifs, sure - Fenrir's sons chase the sun and moon, Achilles chases Hector, No-cha chases Li Ching. But entire stories - that's hard. Yahweh chasing Jonah, maybe?

Two examples come to mind:

Tale Type 313, The Magic Flight features a hero who flees from the ogre king, the father of a princess the hero loves. The two flee from the ogre, and all the while she throws objects behind her that magically turn into obstructions (a comb into a forest, a mirror into a lake, etc.). The story is at the core of Jason and the Argonauts, specifically, the episode about Jason and Medea.

The second example is The Pursuit of Diarmuid and Gráinne. This Irish story is a counterpart of the tale of Lancelot and Guinevere (where the motif also occurs). Diarmuid is the trusted warrior of King Fionn and Gráinne is the young beautiful queen. She falls in love with Diarmuid and compels him to take him on an extensive journey as her husband and his warriors pursue. Sometimes elements of Tale Type 313 appear in association with this ancient Irish tale.

Perhaps - and this is a reach - we see an echo of this in the pursuit of Helen of Troy, which we might regard as the ultimate pursuit story from the ancient world.