V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on the entries of B. We calculate the radius of the neighborhood U . A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304256&_mathId=si1.gif&_user=111111111&_pii=S0024379516304256&_rdoc=1&_issn=00243795&md5=fad62d3e1f0cb8e876b101f46ae96004" title="Click to view the MathML source">Qpclass="mathContainer hidden">class="mathCode"> of p -adic numbers and the field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304256&_mathId=si2.gif&_user=111111111&_pii=S0024379516304256&_rdoc=1&_issn=00243795&md5=5f22dead8ef63b0de358ded7dbce092d" title="Click to view the MathML source">F((T))class="mathContainer hidden">class="mathCode"> of Laurent series over a field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304256&_mathId=si106.gif&_user=111111111&_pii=S0024379516304256&_rdoc=1&_issn=00243795&md5=e16f185adb96f05eb2858c8056f10508" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">.