If a consumer's demand for a good is perfectly inelastic with respect to the price, this means that the consumer is prepared to spend all his available income to that one good, even if this means that he won't consume anything else. This means that the consumer has lexicographic preferences, with this one good above everything else.
Lexicographic preferences are not continuous, and so they cannot be represented by a continuous utility function. They have indifference "curves" that are single points, etc.
So the case of perfectly inelastic demand represents an "exception" at a deeper level than for example Giffen goods represent. Giffen goods are manageable in the context of "usual" preferences, tweaked but not scrapped altogether.
Finally, note that "perfectly inelastic demand" represented as a straight vertical line, does not answer the question "what will happen if, given his budget constraint, the consumer cannot afford to buy the specific quantity $\bar q$?" Will he buy as much as his budget $B$ allows, or will he buy nothing?
If, when he cannot afford the desired quantity $\bar q$, he buys as much as he can, then in reality the demand curve becomes negatively sloped after a price level, and it acquires a "forced" elasticity, dictated by the budget constraint:
The above represents that case of truly lexicographic preferences. In the case where, if the consumer cannot afford $\bar q$, then he buys no quantity of the good, things are more complicated.