I think not; consider pseudosphere, there is no obvious center. Its embedding in 3D space has infinite dimensions but the object itself has finite surface area.

https://en.wikipedia.org/wiki/Pseudosphere

If you mean object with finite dimensions and negative curvature and no center, there seems to be such a thing, see Fig. n066200e:

https://encyclopediaofmath.org/wiki/Negative_curvature,_surf...

EDIT

Rozendorn's idea seems to be still an open question:

https://mathoverflow.net/questions/52851/the-geometry-of-nad...

Rather than think "surface of a balloon inflating", think "surface of an (infinite?) rubber sheet being uniformly stretched". It does intuitively seem like if there is 0 curvature and no "center", then the universe must be infinite though ...