  We report on measurements of the in-plane magnetic penetration depth
($\lambda_{\rm ab}$) in single crystals of Sr$_2$RuO$_4$ down to $\simeq 0.015$
K by means of muon-spin rotation/relaxation. The linear temperature dependence
of $\lambda^{-2}_{\rm ab}$ for $T\lesssim 0.7$ K suggests the presence of nodes
in the superconducting gap. This statement is further substantiated by
observation of the Volovik effect, $i.e.$ the reduction of $\lambda_{ab}^{-2}$
as a function of the applied magnetic field. The experimental zero-field and
zero-temperature value of $\lambda_{\rm ab}=124(3)$ nm agrees with
$\lambda_{\rm ab}\simeq 130$ nm, calculated based on results of electronic
structure measurements reported in [Phys. Rev X 9, 021048 (2019)]. Our analysis
reveals that a simple nodal superconducting energy gap, described by the lowest
possible harmonic of a gap function, does not capture the dependence of
$\lambda_{\rm ab}^{-2}$ on $T$, so the higher angular harmonics of the energy
gap function need to be introduced.
