<p style="text-align:justify;"> We extend Urban’s construction of eigenvarieties for reductive groups <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> such that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℝ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math> has discrete series to include characteristic <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> -analytic manifold taking values in a complete Tate <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>p</mi></mrow></msub></math> -algebra in which <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> -adic Lie groups given by Johansson and Newton. </p>