[Alcuin]

• When Aragorn and Frodo watched the battle between Gandalf and the Nazgûl on Weathertop from the Midgewater Marshes, Aragorn said that, “It is like lightning that leaps up from the hill-tops,” and Frodo could see “white flashes” in the distance.
• This sounds very like the “stab of white light” the Company of the Ring saw at the top of the stair when the Chamber of Mazarbul collapsed on the Balrog as it tried to wrench the door from Gandalf’s control.
• When Gandalf was wielding Glamdring against the flaming red sword of the Balrog, there was a “stab of white fire”. (Aside: there were flames running down the sword of the Lord of the Nazgûl when he confronted Gandalf at the Gates of Minas Tirith.)
• When Gandalf broke the Durin’s Bridge, there was “a blinding sheet of white flame”.
• And when Gandalf chased off the pursuing winged Nazgûl to rescue Faramir and his companions as they retreated to Minas Tirith just before the beginning of the siege, Gandalf “raised his hand, and from it a shaft of white light stabbed upwards.”

All of these sound like lightening. At the very least, they are all some kind of intense, white light that the Nazgûl and the Balrog intensely disliked. And as you point out, at least on the peak of Zirakzigil, Gandalf did not have a staff with him.

• In TT, “King of the Golden Hall”, there is the episode where Gandalf silences Wormtongue’s deceits:

  “...I have not passed through fire and death to bandy crooked words with a serving-man till the lightning falls.” He raised his staff. There was a roll of thunder. The sunlight was blotted out from the eastern windows; the whole hall became suddenly dark as night. The fire faded to sullen embers. Only Gandalf could be seen, standing white and tall before the blackened hearth.
  
  In the gloom they heard the hiss of Wormtongue’s voice: “Did I not counsel you, lord, to forbid his staff? That fool, Hama, has betrayed us!” There was a flash as if lightning had cloven the roof. Then all was silent. Wormtongue sprawled on his face. …

  I had always assumed that Gandalf produced the lightning using his staff; now I am not so sure.

Gandalf did use the staff when starting the fire on Caradhras, and his staff broke when he broke Durin’s Bridge, as doug*platypus has pointed out. But perhaps the bridge incident can be explained another way: “The Istari” essay in UT says early on that the Istari

…were forbidden to reveal themselves in forms of majesty, or to … open display of power, but coming in shapes weak and humble were bidden to advise and persuade Men and Elves to good…

When Gandalf fought the Balrog in Moria, although he was still confined to a “shape weak and humble”, it was a battle between two Maia, something that had not taken place in Middle-earth since Eönwë the Herald of Manwë fought in the War of Wrath. Perhaps in breaking his staff, Gandalf was signifying that he was putting off any pretense of appearing to be a Man or Elf, summoning all the strength available to him as Olórin the Maia, hobbled as he might remain by the restrictions imposed upon him when he left Valinor. That was his best hope of saving the Ring-bearer and the Heir of Elendil from Durin’s Bane; and of course, he died after overthrowing the Balrog.

Which leads to this: If that last paragraph is correct, and Gandalf broke his staff as a sign that he had broken his restrictions as an Istar, did he die merely from physical injuries sustained fighting the Balrog, or also because that was the natural outcome of his decision to “fight like a Maia”?