The union said it had put forward several options to the airline but no agreement could be reached. British Airways has branded the decision to take industrial action as "completely unacceptable".
Pilots will strike on September 9, 10 and 27. The airline estimates that "tens of thousands" of passengers will be affected.
In a statement, Balpa said: "Over several days of ACAS talks BALPA put forward a number of packages that we believe would have resolved this dispute without a strike, and which we could have recommended to our members for acceptance prior to strike action. BA did not accept any of these packages, and it is clear following discussions with members over the last few days that BA's most recent offer will not gain the support of anywhere near a majority of its pilots.
"In these circumstances, with a 93% vote in favour of taking industrial action, and with no prospect of any further meaningful talks, we have no choice but to call this action.
"A day of strike action will cost BA around £40 million. Three days will cost in the region of £120 million. The gap between BA's position and BALPA's position is about £5m. Our proposal remains on the table should BA wish to reach agreement prior to strike action.
"British Airways is an extremely profitable and successful company, and pilots have been proud to play their part in that. In 2018 the company announced profits of £2 billion. Over recent years BA pilots have made sacrifice after sacrifice to assist the company such as taking a pay cut, productivity increases, closing the final salary pension scheme, giving up annual leave days, a new rostering system, and reducing flying pay.
"In what is British Airways' centenary year, this will be the very first time its pilots will go on strike. They do so as a last resort and with enormous frustration at the way the business is now being run.
"Our ballot is valid until January, and more dates may be announced until such time as this matter is resolved."
In response to the announcement of strike dates by BA pilots, the airline said: "It is completely unacceptable that Balpa is destroying the travel plans of tens of thousands of our customers with this unjustifiable strike action.
"We are extremely sorry that after many months of negotiations, based on a very fair offer, Balpa has decided on this reckless course of action.
"We are now making changes to our schedule. We will do everything we can to get as many people away on their journeys as possible.
"However, it is likely that many of our customers will not be able to travel and we will be offering refunds and re-bookings for passengers booked on cancelled flights."