<p>In this work, a set-valued minimax programming problem (shortened to SVMP) is taken into account. In the broad sense of higher-order arcwisely connected set-valued maps, we introduce the concept of higher-order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-arcwisely connectedness of set-valued maps (abbreviated as SVMs). Under higher-order contingent epidifferentiation and higher-order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-arcwisely connectivity suppositions, the higher-order sufficient criteria for Karush-Kuhn-Tucker (KKT) optimality are constructed for the problem (<InternalRef RefID="Equ2">MP</InternalRef>). Additionally, we develop the higher-order Mond-Weir (<i>MWD</i>), Wolfe (<i>WD</i>), and mixed (<i>MD</i>) kinds of duality and demonstrate the higher-order strong, weak, and converse theorems of duality among our main problem (<InternalRef RefID="Equ2">MP</InternalRef>) and the associated duals under higher-order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-arcwisely connectivity supposition.</p>