We study property (T) and the fixed point property for actions on L^p and other Banach spaces. We show that property (T) holds when L² is replaced by L^p (and even a subspace/quotient of L^p), and that in fact it is independent of  1 ≤ p < ∞. We show that the fixed point property for L^p follows from property (T) when  1 < p <  2+ε. For simple Lie groups and their lattices, we prove that the fixed point property for L^p holds for any  1 < p < ∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.