  We measure isotope shifts for five Yb$^+$ isotopes with zero nuclear spin on
two narrow optical quadrupole transitions ${}^2S_{1/2} \rightarrow
{}^2D_{3/2}$, ${}^2S_{1/2} \rightarrow {}^2D_{5/2}$ with an accuracy of $\sim
300$ Hz. The corresponding King plot shows a $3 \times 10^{-7}$ deviation from
linearity at the 3 $\sigma$ uncertainty level. Such a nonlinearity can indicate
physics beyond the Standard Model (SM) in the form of a new bosonic force
carrier, or arise from higher-order nuclear effects within the SM. We identify
the quadratic field shift as a possible contributor to the nonlinearity at the
observed scale, and show how the nonlinearity pattern can be used in future,
more accurate measurements to separate a new-boson signal from nuclear effects.
