  A major challenge in quantum metrology is the generation of entangled states
with macroscopic atom number. Here, we demonstrate experimentally that atomic
squeezing generated via non-linear dynamics in Bose Einstein condensates,
combined with suitable trap geometries, allows scaling to large ensemble sizes.
We achieve a suppression of fluctuations by 5.3(5) dB for 12300 particles,
which implies that similar squeezing can be achieved for more than 10$^7$
atoms. With this resource, we demonstrate quantum-enhanced magnetometry by
swapping the squeezed state to magnetically sensitive hyperfine levels that
have negligible nonlinearity. We find a quantum-enhanced single-shot
sensitivity of 310(47) pT for static magnetic fields in a probe volume as small
as 90 $\mu$m$^3$.
