browserAction.disable()

Disables the browser action for a tab, meaning that it cannot be clicked when that tab is active.

tabIdOptional

integer. The id of the tab for which you want to disable the browser action.

Disable the browser action when clicked, and re-enable it every time a new tab is opened:

browser.tabs.onCreated.addListener(() => {
  browser.browserAction.enable();
});

browser.browserAction.onClicked.addListener(() => {
  browser.browserAction.disable();
});

Disable the browser action only for the active tab:

browser.browserAction.onClicked.addListener((tab) => {
  browser.browserAction.disable(tab.id);
});