Could matter be infinite?
Yes.  Currently it is at least plausible that the universe is spatially flat.  If that is the case then, barring global topological weirdness, it is also spatially infinite.  That would mean it contains an infinite amount of matter.
Disclaimer. I don't know how large the uncertainties on spatial flatness currently are: if Wikipedia is to be trusted $\Omega = 1.00 \pm 0.02$ ($1$ being flat) based on data from WMAP and Planck.