  We prove finiteness and base change properties for analytic cohomology of
families of $L$-analytic $(\varphi_L,\Gamma_L)$-modules parametrised by
affinoid algebras in the sense of Tate. For technical reasons we work over a
field $K$ containing a period of the Lubin-Tate group, which allows us to
describe analytic cohomology in terms of an explicit generalized Herr complex.
