The units of rate constant and rate ofreaction are identical for

$\mathrm{nA}\to$Product

rate $=\mathrm{K}[\mathrm{A}{]}^{\mathrm{n}}=\mathrm{K}[\mathrm{A}{]}^0=\mathrm{K}$

Only when n is zero is it possible for the rate to be equal to K.
For a zero order reaction, the units for the rate constant and reaction speed are ${\text{ mol.lit }}^{-1}{\sec }^{-1}$.