When I'm without a solid carbide drill bit that fits the bill, I've put holes in a few things using a hand-sharpened carbide-tipped masonry drill bit. They're easy to find and cheap. Have a look at the tip before you buy it and try to picture whether you can sharpen it or not. Some won't work as well as others.
Sharpen it using a silicon carbide or diamond wheel with minimal relief so it doesn't dig in too aggressively. You won't get anything resembling spiral chips, more like saw dust or just plain dust. If you're drilling through solid then maybe try to punch or nick a starting hole to keep the drill from walking. Drilling out an existing hole will probably be harder because it will want to chatter like crazy.
Clamp everything rigidly, turn slow, feed light, and keep it all cool.
If it doesn't work, you've only lots a few bucks (or pounds, euro, quid.... whatever). You may need more than one...