The Parliament of the United Kingdom defines the noun ping-pong as follows:

‘Ping-pong’ refers to the to and fro of amendments to Bills between the House of Commons and the House of Lords.

This noun occurs, for example, in the following from UK government in talks with charter airline over Rwanda deportation flights, by Rajeev Syal, home affairs editor, published in The Guardian (London and Manchester, England) of Friday 12^th April 2024:

No 10 prepares for the latest Rwanda bill to return to parliament on Monday in its attempt to deter asylum seekers from travelling across the Channel in small boats.
Government insiders remain confident that the bill will pass by the end of April after another round of parliamentary ping pong between the House of Commons and the House of Lords, and that flights will take off in the spring.

However, originally, in 1902, the noun ping-pong was used of British parliamentary debates in reference to rapid verbal exchanges between two parties.

Amusingly, this coincided with the craze for the recently-invented game of ping-pong—to the extent that some Members of Parliament asked that this game be installed in the Palace of Westminster.

The following are some of the earliest occurrences that I have found of the original political use of the noun ping-pong—and of those MPs’ craze for the game:

1-: From The Yorkshire Post (Leeds, Yorkshire, England) of Friday 24^th January 1902 [No. 17,031, page 7, column 1]:

PARLIAMENTARY SKETCH.
(From our Special Correspondent.)

Westminster, Thursday Night.
The Celtic fringe has been flapping the night through. When the Irish have made up their minds to have an evening of it they do it well, with noise and bluster and Hibernian hullabaloo. Mr. John Redmond sat for an hour up in his corner seat, deep meditation on his Napoleonic features, and his right hand stuck deep within his unbuttoned vest. He would throw the thunderbolt at the Saxon Government for callousness towards the ever-pressing Irish land question. It was to be a great speech. So for just sixty minutes he sat wrapped in reverie, whilst around his henchmen and hewers and carriers were shouting and barking and holding high revelry during the time set apart for Ministers to stand up and let who will shy questions at them.
The House was full, as it always is in the heckling hour. The lights were up, for the afternoon was grey and dim, and there was the bustle of business. Getting on for a hundred questions were fired at the Treasury Bench, and sharp rallies in Parliamentary ping-pong took place across the table.

2-: From Pictures in Parliament, by Ralph Rambleton, published in The Hackney Express and Shoreditch Observer (London, England) of Saturday 25^th January 1902 [Vol. 46, No. 2,352, page 3, column 1]:

I have been looking at the debates on the King’s Speech, and have watched, with a sad, sad smile, the Liberal party’s amendment. Why is it necessary to forge at such pains an amendment at all? When will a genius arise in the Opposition ranks who will discover the elemental principle that for an Opposition to ride for a fall is political insanity? If the war is not the business of Liberals, why not let it alone? That would be dignified, and would at least lessen the score of the government in this Parliamentary Ping Pong.

3-: From Parliamentary Notes, published in The Daily Telegraph (London, England) of Tuesday 28^th January 1902 [No. 14,581, page 9, column 3]:

Ought ping-pong to be tolerated within the sacred precincts of the Palace of Westminster? This is the question which is at present agitating the minds of members of Parliament, who, having seen how much pleasure can be derived from the game elsewhere, perceive in it a means of lessening the tedium of life in the House of Commons, where chess, draughts, and smoking are about the only relaxations permitted to weary legislators. It is feared that the First Commissioner of Works and the Serjeant-at-Arms would stand aghast at the spectacle of grave and reverend politicians bounding about the table in frantic efforts to hit the little balls. Moreover, the problem of accommodation presents a difficulty. But the enthusiastic advocates of ping-pong are sanguine that before long they will induce the authorities to listen to their views.

4-: From Chat of Old London, published in The Sun (Baltimore, Maryland, USA) of Sunday 2^nd February 1902 [Vol. 130, No. 78, page 10, column 1]:

Ping-Pong Invades Parliament.

Notwithstanding the talk in many quarters that ping-pong, or table tennis, is sure to die out soon, like every other fad, the fact remains that the game has a firm grip on the popular fancy just now. It has even got into Parliament, though not before the House, of course.
A number of members of the House of Commons are seriously considering the advisability of having the game permanently installed in one of the committee rooms as a means of beguiling some of the frequently tedious hours spent in listening to long, set speeches. Whether the ping-pong enthusiasts will have their way remains to be seen.
It is amusing to notice that the golf widow has been superseded by the ping-pong widow. The former used to see her spouse in the evening, but not so the latter. After swallowing a hasty dinner he starts straight for his ping-pong club with a racquet under his arm and a wild look in his eyes. He says nothing during the meal, his sole conversation being fragmentary remarks which he makes in his sleep about “smashes” and “half volleys.”

5-: From Notes at Westminster, by J. Foster Fraser, published in the Aberdeen Daily Journal (Aberdeen, Aberdeenshire, Scotland) of Tuesday 18^th February 1902 [No. 14,688, page 5, column 5]:

After a game of Parliamentary ping-pong across the table between the leaders of the chief parties, the consideration of the new motion was held over till Thursday.

6-: From Notes on Pastimes, by ‘The Critic’, published in the Sheffield Daily Telegraph (Sheffield, Yorkshire, England) of Wednesday 2^nd April 1902 [No. 14,578, page 10, column 1]:

Ping-Pong in Parliament.
Nothing is sacred to ping-pong, and its introduction into Westminster was merely a matter of time. It is not yet vouchsafed us to see Mr. Chamberlain and Mr. Dillon hurling xylonite balls across the room at each other instead of epithets and invective, or Mr. Balfour and Sir H. Campbell-Bannerman proclaiming the gospel of “love all,” but that may come. Meanwhile it is sufficient to know that in a private room, set apart for the use of the reporters of one of the London papers, the journalists have found a new use for their note-books, which now act as rackets to propel balls over an improvised net. This is the thin edge of the wedge, and, as one Member now employs his spare time in knitting, is it too much to expect that legislators will soon be devoting the ’tween-division periods to a pastime they no doubt pursue in other places?

7-: From a transcript of the speech that Principal Marshall Lang delivered during the dinner given by Lord Provost Fleming in the Town and County Hall, Aberdeen, on Wednesday 9^th April 1902—transcript published in the Aberdeen Daily Journal (Aberdeen, Aberdeenshire, Scotland) of Thursday 10^th April 1902 [No. 14,732, page 6, column 6]:

I take it that there is not a more high-minded body of men than that which gathers together in Westminster—(cheers)—but just because we believe that we must discountenance the use of intemperate language. (Hear, hear, and cheers.) Parliament should not represent the Hooliganism of the country—(hear, hear)—but should represent all that is educated and refined. There may be a kind of ping pong in Parliament, a throwing of the balls of wit and retort from one side of the House to the other, but I think we are all agreed that we should demand of our legislators that they do not discredit Parliament—(hear, hear)—by language that would not be tolerated in any assembly, and I think we expect that our legislators, whether in Parliament or out of Parliament, will never soil with ignoble use or with ignoble speech the grand old name of gentleman. (Hear, hear, and cheers.)

The earliest occurrence that I have found of the noun ping-pong used in reference to the to and fro of amendments to bills between two Houses of Parliament is not British, but Australian—it is from Figaro’s Latest, published in The Queensland Figaro (Brisbane, Queensland, Australia) of Thursday 23^rd October 1902 [page 14, column 1]—the Legislative Council was the Upper House of the Queensland Parliament:

The gentlemen of the Legislative Council seem determined to harass the Income Tax Bill. The probability is that on its return to the Assembly it will be so altered by amendments that the Speaker will rule it out of order. Then will commence an undignified game of Parliamentary ping-pong between the two Houses, the Bill being the ball.