Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve

New version ; modified Introduction and Section 6We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub