The penetration rate of power-law non-Newtonian liquids in a circular tube driven by the capillary pressure is derived. The dimensionless fractional height of liquid in the tube relative to the equilibrium height, chi = h(t)/hd(infinity), is calculated as a function of an appropriately chosen dimensionless time, permitting expression of the height-time dependence as a function of the reciprocal of the exponent in the power-law model, s = 1/n. It is found that the rate of penetration and, therefore, also withdrawal of liquid is dramatically retarded as the liquid becomes more strongly shear-thinning, i.e., as n becomes small relative to unity.