Step 1: Base rule under Article 356.
President’s Rule (failure of constitutional machinery in a state) is proclaimed for six months at a time.
Step 2: Extensions and upper cap.
It can be extended every six months, but not beyond three years, and only if {both} conditions hold:
(i) a National Emergency is in operation in the whole or any part of India; and
(ii) the Election Commission certifies that elections cannot be held in the concerned state.
Step 3: Conclude.
\[
\boxed{\text{Maximum permissible extension = 3 years (Option C).}}
\]