  This article is the first one of a series of three articles devoted to direct
images of isocrystals: here we consider isocrystals without Frobenius
structure; in the second one (resp. the third one), we will introduce a
Frobenius structure in the convergent (resp. overconvergent) context. For a
liftable proper smooth morphism we establish the overconvergence of direct
images, owing to a base change theorem for a proper morphism between rigid
analytic spaces. This result partially answers a conjecture of Berthelot on the
overconvergence of direct images under a proper smooth morphism.
