Does Batman have a spacesuit?

He has at least nine (and an unclear one, and a bonus one).

Ordered by media and publication date.

Seven in the comics

• One shown in tilley31's answer (1969).

• The unclear one would be the spacesuit he wore in Justice League of America #202 (1982). It looks like a regular space suit, but it has a Bat-Symbol on it and what looks like a utility belt which, at the very least, contains tools to repair the JLA satellite. There's not enough material to state it was made by Batman, and the comic only vaguely refers to it as "Batman's spacesuit":

  

• One worn in Batman/Superman #64 (2009), while Batman was "casually" inspecting a shipwreck in space. Despite the red symbol, it has nothing to do with Batman Beyond.

  

• One worn by Dick Grayson (who was going around by "Batman" at that time) in Justice League of America #55 (2011):

  

• Another shown in Batman/Superman #28 (2015); by the way Alfred words it, the thing was certainly built by Batman.

  SUPERMAN: There's something you... There's something Batman needs to see.
  ALFRED: I'll prepare your suit, sir.
  BATMAN: I'm guessing, Alfred, given recent events, whatever​ Superman wants to show me is on the Moon.
  ALFRED: I'll prepare your other suit, sir.

  

  

• Another shown in DC Cybernetic Summer (2020). Destroyed along with Brother Eye's drones in a satellite explosion.

  

• Another very briefly shown in Detective Comics #1027 (2020), used to fight against Brainiac.

  

One LEGO Batman Space Batsuit

The impressive Bat-Space shuttle building kit, set in The LEGO Batman Movie universe, features a "Space Batsuit":

One in Batman: The Brave and the Bold

Seen in the very first episode, "The Rise of the Blue Beetle" (2008): a seemingly regular Batman suit, combined with a breathing thingy and a jetpack. I don't know if that counts as a space suit, but he's flying in outer space and not dying. I don't think kevlar and leather allow for that.

One in LEGO Batman 3: Beyond Gotham

The minifigure sold by LEGO on Amazon is listed as "Space Batman". Batman can be seen in action wearing this suit in various videos of the game.

Bonus

As a bonus, here's a Batman spacesuit which is supposedly not a Batman spacesuit. It's featured in Superman/Batman #78 (2011), where two kids imagine how a "Batman vs Superman" scenario would play out. Part of it involves Superman taking Batman to space, and Batman revealing his suit actually contains an unfolding spacesuit inside it. And while it may just be a kid's imagination, at the end of the issue, Batman (who was watching from above) tells Superman that the kid had "a lot of it right, how I'd beat you". Without detailing exactly what...

End note: because he's Batman

Standard comic rules apply to the whole DC universe, and Batman adds another layer to that. If the writer decides his regular suit + breathing device is enough to last him two whole minutes in outer space, then it will be, as shown in Legends of the Dark Knight #2 (2011). Because he's Batman!

_{Hat tip to TheLethalCarrot for locating that panel's issue for me!}