〜が早いか
no sooner had … than
Verb dictionary / た-form + が早いか
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が早いか says that the second action happened the very instant the first one did, almost at the same time: ベルが鳴るが早いか、生徒たちは教室を飛び出した, "no sooner had the bell rung than the students burst out of the classroom".
The second half is a past event that someone observed, usually sudden and energetic. You don't use it for your own plans, requests or intentions.
It's a literary cousin of たとたん (N3) and か〜ないかのうちに (N2). や否や and なり (both N1) are very close. The feeling is "you couldn't even tell which came first".
Compare 次第 (N2), "as soon as", which is about future plans: 着き次第連絡します. が早いか only reports what already happened.
Examples
ベルが鳴るが早いか、生徒たちは教室を飛び出した。
No sooner had the bell rung than the students burst out of the classroom.
子どもは家に帰るが早いか、ゲームを始めた。
The moment the kids got home, they started playing video games.
彼は給料をもらうが早いか、全部使ってしまった。
No sooner had he been paid than he spent it all.
店が開くが早いか、客が殺到した。
The moment the shop opened, customers came flooding in.
犬は餌を見るが早いか、飛びついた。
The dog pounced on its food the instant it saw it.