For the verb meaning (1) to beat or (2) to scold or berate, lambaste is the preferred spelling in American and Canadian English, while lambast is preferred in varieties of English from outside North America.
While the exact derivation of the word is not definitively known, the OED posits that it’s a combination of lam and baste, both of which bear the sense (now archaic for both words) to beat soundly. Other sources agree. And lambaste is the older form. In historical Google Books searches, lambast is almost nonexistent before 1850. It appears increasingly in the second half of that century, including in works by Charles Dickens and Rudyard Kipling (British authors).
That’s the way of these sharks; they first hand-cuff a man’s ideas with hard words, then lambaste him for not following suit; and afterwards make him pay the piper. [A [Green Hand’s First Cruise, by Josiah Cobb (1841)]
“For,” he said, “if you lambast the critters, it is a fact, they’ll drown theirselves just to spite you.” [All the Year Round, by Charles Dickens (1866)]
“And if they don’t cry—I repeat, if they don’t cry, I’ll lambaste the stuffing out of them.” [Smoke Bellow, by Jack London (1912)]
Browns Lambast Yanks 10-2 in Series Opener [Bonham Daily Favorite (1943)]
Dear Abby: I didn’t appreciate that letter from Elaine lambasting people who send annual newsletters to their friends at Christmas time. [Evening Independent (1969)]
Tagg Romney, Mitt Romney’s eldest son, took to Twitter on Monday to lambast the “liberal media” for “mocking my dead grandpa”. [Telegraph (2012)]
Since then, the two leaders have maintained a personally amicable relationship but have continued to lambaste each other’s policies. [Wall Street Journal (2012)]
This ngram graphs the use of lambast and lambaste in a large number American books published from 1850 to 2019:
And this ngram charts the same in British books: