  We initiate the study of exotic Dehn twists along 3-manifolds $\neq S^3$
inside $4$-manifolds, which produces the first known examples of exotic
diffeomorphisms of contractible 4-manifolds, more generally of definite
4-manifolds, and exotic diffeomorphisms of 4-manifolds with $\neq S^3$ boundary
that survive after one stabilization. We also construct the smallest closed
4-manifold known to support an exotic diffeomorphism. These exotic
diffeomorphisms are the Dehn twists along certain Seifert fibered 3-manifolds.
As a consequence, we get loops of diffeomorphisms of 3-manifolds that
topologically extend to some 4-manifolds $X$ but not smoothly so, implying the
non-surjectivity of $\pi_1(\mathrm{Diff}(X)) \to \pi_1(\mathrm{Homeo}(X))$. Our
method uses $2$-parameter families Seiberg-Witten theory over $\mathbb{RP}^2$,
while known methods to detect exotic diffeomorphisms used $1$-parameter
families gauge-theoretic invariants. Using a similar strategy, we construct a
new kind of exotic diffeomorphisms of 4-manifolds, given as commutators of
diffeomorphisms.
