We use droplet microfluidics to produce monodisperse elastomeric microbubbles consisting of gas encapsulated in a polydimethylsiloxane shell. These microbubbles withstand large, repeated deformations without rupture. We perform mu N-scale compression tests on individual microbubbles and find their response to be highly dependent on the shell permeability; during deformation, the pressure inside impermeable microbubbles increases, resulting in an exponential increase in the applied force. Finite element models are used to interpret and extend these experimental results enabling the design and development of deformable microbubbles with a predictable mechanical response. Such microbubbles can be designed to repeatedly transit through the narrow constrictions found in a porous medium functioning as probes of the local pressure.