We've all done it. It is so very easy to hurriedly type with or without that little apostrophe and it is so very easy to overlook that you've done so.
It doesn't help that nouns (person, place, or thing, see Schoolhouse Rock) are made possessive with an apostrophe: Terry's report, the city's traffic, the project's deadline.
The correct answer is "a) Late? It's okay; its deadline was extended."
Refresher on the "why":
Pronouns (take the place of nouns, see another Schoolhouse Rock video) do not use apostrophes to be possessive. "its deadline," "the report is hers," "his coffee."
Contractions (shoot, there's no Schoolhouse Rock video for this one!) use an apostrophe to indicate a missing letter, such as "it is" turning into "it's" or "do not" becoming "don't."
Take a moment after typing to re-read and mentally substitute "it is" for any its/it's instances to make sure your autopilot was set correctly. I know mine occasionally misfires.