  The superconducting state of an optimally doped single crystal of
Ba(Fe$_{0.93}$Co$_{0.07}$)$_2$As$_2$ was investigated by $^{75}$As NMR in high
magnetic fields from 6.4 T to 28 T. It was found that the Knight shift is least
affected by vortex supercurrents in high magnetic fields, $H>11$ T, revealing
slow, possibly higher order than linear, increase with temperature at $T
\lesssim 0.5 \, T_c$, with $T_c \approx 23 \, K$. This is consistent with the
extended s-wave state with $A_{1g}$ symmetry but the precise details of the gap
structure are harder to resolve. Measurements of the NMR spin-spin relaxation
time, $T_2$, indicate a strong indirect exchange interaction at all
temperatures. Below the superconducting transition temperature vortex dynamics
lead to an anomalous dip in $T_2$ at the vortex freezing transition from which
we obtain the vortex phase diagram up to $H = 28$ T.
