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We construct a self-interacting dark matter model that could simultaneously explain the observed muon anomalous magnetic moment. It is based on a gauged <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> <mo>&#x2212;<!-- - --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03C4;<!-- τ --></mi> </mrow> </msub> </mrow> </msub> </math> extension of the standard model, where we introduce a vector-like pair of fermions as the dark matter candidate and a new Higgs boson to break the symmetry. The new gauge boson has a sizable contribution to muon (g - 2), while being consistent with other experimental constraints. The <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> <mo>&#x2212;<!-- - --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03C4;<!-- τ --></mi> </mrow> </msub> </mrow> </msub> </math> Higgs boson acts as a light force carrier, mediating dark matter self-interactions with a velocity-dependent cross section. It is large enough in galaxies to thermalize the inner halo and explain the diverse rotation curves and diminishes towards galaxy clusters. Since the light mediator dominantly decays to the <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> <mo>&#x2212;<!-- - --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03C4;<!-- τ --></mi> </mrow> </msub> </mrow> </msub> </math> gauge boson and neutrinos, the astrophysical and cosmological constraints are weak. We study the thermal evolution of the model in the early Universe and derive a lower bound on the gauge boson mass.