The phrase "a ball at which" is correct and usable in written English.
It can be used in contexts where you are describing an event or activity involving a ball, often in sports or games.
Example: "He is known for being a skilled player, especially at a ball at which he can showcase his talents."
Alternatives: "a game in which" or "an event where".
Once a year, for example, Arpad Busson, a French financier based in London, gives a ball at which rich folk outbid each other for donated extravagances.
Then a number of the Prince's retinue announce that the Prince himself will shortly arrive and invite Don Magnifico and his daughters to a ball at which he will choose his future wife.
This introduced the "outsider" character that became a hallmark of his ballets, in this case a female clown who attends a ball at which her host falls in love with her until she loses the mask that has made her attractive.
After the war, Hara became a key face in Japan's cinematic revival, and was cast in a lead role in Akira Kurosawa's No Regrets for Our Youth (1946) and Kōzaburō Yoshimura's A Ball at the Anjo House (1947), both of which lamented the price Japan paid for its prewar militarism.
Darcy surprises Elizabeth with a dance offer at a ball at Netherfield, which she grudgingly but politely accepts.
The batter also can reach first base if a pitched ball at which he does not swing strikes any part of his person.
On his return he attends a glitzy ball at which he recognises the glamorous hostess as Tatyana, now married to a general, and realises that he has always loved her.
"They gave me a ball at New York, at which Three Thousand people were present — and a public dinner besides".
England's second opportunity came when Lumb reached a near run-a-ball century at which point England, going well, required 99 from 15 overs with eight wickets in hand.
The day consists of a church ceremony, during which the girl is blessed, and then a dinner or ball, at which she has the first dance with her father.
Still more startlingly, the domestics later host a party – the Servants' Ball – at which they dance with their employers.
