  Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly
isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property
$({\BP}_0^V)$, for $V$ be a Banach space: a locally compact group $G$ has
property $({\BP}_0^V)$ if every affine isometric action of $G$ on $V$, such
that the linear part is a $C_0$-representation of $G$, either has a fixed point
or is metrically proper. We prove that solvable groups, connected Lie groups,
and linear algebraic groups over a local field of characteristic zero, have
property $({\BP}_0^V)$. As a consequence for unitary representations, we
characterize those groups in the latter classes for which the first cohomology
with respect to the left regular representation on $L^2(G)$ is non-zero; and we
characterize uniform lattices in those groups for which the first $L^2$-Betti
number is non-zero.
