Drama is a very culture-specific thing. It's hard to make clear exactly how specific it is. Drama isn't a standard part-and-parcel aspect of cultures generally: in principle there's no reason to expect a given culture to have drama. We talk about different forms of ancient drama under a single heading, "drama", but if they have independent origins, they can only be lumped together with violence.

With that in mind, we can talk about (a) things that resemble drama to some extent; and (b) things that have a genetic relationship to a more familiar form of drama.

Category (a) is hard to talk about: a lot will depend on what we agree "drama" means. Any quibble over semantics is enough to make the discussion collapse. Certainly there are pieces of literature that come closer to looking "drama-ish" than others, especially the book of Job in the Hebrew Bible.

But is "drama" any form where someone adopts the persona of another person? That's probably a bit too loose: on that definition, any storyteller who tells a pre-existing story is doing a one-person show where they're playing the part of the original storyteller. Do there have to be multiple humans playing roles? That's too strict: that would mean a one-person show can't be a drama. Does it have to be humans? That would rule out puppet theater; in modern media, it would rule out animation. Does it have to be "imitative" rather than "diegetic" (as per Plato's definition)? That would be a ban on reported narrative within the drama (any occasion where a character relates something that happened "offstage"). It boils down to how you envisage the relationship between the speaking performer and the narrator, so this is a messy topic.

Category (b) is a lot easier, because we don't have to wiffle about definitions of what exactly we're talking about.

It so happens we do have one example of a drama, Ezekiel of Alexandria's Exagoge, which closely follows the generic patterns of Greek tragedy: it's in Greek, in iambic trimeter, and so on. The story is that of the Exodus. It's not intact - 269 lines survive - and its date is uncertain: it was probably written sometime between the 3rd and 1st centuries BCE, but exactly when is heavily debated. Moses is the main character. Remarkably, God appears as a character too, in the burning bush scene. Here's an excerpt (lines 120-131, trans. Jacobson):

God: What is that in your hands? Speak quickly.
Moses: A rod wherewith to chastise beasts and men.
God: Throw it on the ground and withdraw quickly.
For it shall turn into a fearsome snake and you will marvel at it.
Moses: There, I have thrown it down. Oh Master, be merciful.
How dreadful, how monstrous. Have pity on me.
I shudder at the sight, my limbs tremble.
God: Have no fear. Reach out your hand and seize its
tail. It shall turn back into a rod.
Now put your hand into your bosom and with­ draw it.
Moses: There, I've done it. It's become like snow.
God: Put it back into your bosom and it shall be as it was before.

Here, the semantic argument argument is whether we should call it "Israelite/Hebrew" drama, because obviously it's not in Hebrew and it wasn't written in Israel. It's not at all clear whether Ezekiel actually knew Hebrew: Jacobson's edition discusses this at pages 40-47. Ezekiel made heavy use of Biblical texts, but consistently drew on the Septuagint rather than the Hebrew text; at the same time, there are a number of linguistic features that look Semitic in flavor (like the use of the future tense in place of the imperative, Greek kai being used like Hebrew vav, etc.).

So as I see it, these are the two main avenues to answering this question: (a) quibbling over what "drama" means, or (b) pointing to an example that imitates a more familiar model of "drama".

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Edit: I see from Jacobson that there's no actual evidence that Ezekiel was Alexandrian: that's just the usual assumption because, as he puts it, "It is reasonable to ask whether any other city in the Hellenistic world would really have suited a Greek play on a Jewish subject."