  Long ago, Fontaine formulated conjectures (now theorems) relating \'etale and
de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier
work we have shown that pro-\'etale and de Rham cohomologies of analytic
varieties in the two extreme cases: proper and Stein, are also related. In the
proper case, the comparison theorems are similar to those for algebraic
varieties, but for Stein varieties they are quite different.
  In this paper, we state analogs of Fontaine's conjectures for general smooth
dagger varieties, that interpolate between these two extreme cases, and we
prove them for many ``small'' varieties (cases we deal with include products of
overconvergent affinoids and proper varieties, analytifications of algebraic
varieties, or "almost proper" dagger varieties). The proofs use a
``geometrization'' of all involved cohomologies in terms of quasi-Banach-Colmez
spaces (qBC's for short, quasi- because we relax the finiteness conditions).
The heart of the proof relies on delicate properties of BC's and qBC's. These
properties should be of independent interest and we have devoted a large part
of the paper to them.
