The smallest polyomino with a hole is the heptomino in the form of a $3 \times 3$
square with the center $1 \times 1$ square and a corner $1 \times 1$ square missing:
From 35 Hexominoes and by inspection, none of the $35$ hexominoes has a hole.
From Number of Heptominoes and by inspection, exactly one of the $108$ heptominoes has a hole.
This is the smallest polyomino with a hole.