We study a mixed boundary value problem for the p</mi></math>" id="MathJax-Element-4-Frame" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0">p-Laplace equation <msub is="true"><mrow is="true"><mi is="true">&#x394;</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msub><mi is="true">u</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">0</mn></mrow></math>" id="MathJax-Element-5-Frame" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0">Δpu=0 in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. Existence of weak solutions to the mixed problem is proved both for Sobolev and for continuous data on the Dirichlet part of the boundary. We also obtain a boundary regularity result for the point at infinity in terms of a variational capacity adapted to the cylinder.