diff --git a/base/reducedim.jl b/base/reducedim.jl
index f5a22940310cf..21dff3b46ab37 100644
--- a/base/reducedim.jl
+++ b/base/reducedim.jl
@@ -146,16 +146,18 @@ for (f1, f2, initval, typeextreme) in ((:min, :max, :Inf, :typemax), (:max, :min
             T = _realtype(f, promote_union(eltype(A)))
             Tr = v0 isa T ? T : typeof(v0)
 
-            # but NaNs and missing need to be avoided as initial values
+            # but NaNs, missing and unordered values need to be avoided as initial values
             if v0 isa Number && isnan(v0)
                 # v0 is NaN
                 v0 = oftype(v0, $initval)
             elseif isunordered(v0)
                 # v0 is missing or a third-party unordered value
                 Tnm = nonmissingtype(Tr)
-                # TODO: Some types, like BigInt, don't support typemin/typemax.
-                # So a Matrix{Union{BigInt, Missing}} can still error here.
-                v0 = $typeextreme(Tnm)
+                if Tnm <: Union{BitInteger, IEEEFloat, BigFloat}
+                    v0 = $typeextreme(Tnm)
+                elseif !all(isunordered, A1)
+                    v0 = mapreduce(f, $f2, Iterators.filter(!isunordered, A1))
+                end
             end
             # v0 may have changed type.
             Tr = v0 isa T ? T : typeof(v0)
@@ -186,12 +188,18 @@ function reducedim_init(f::ExtremaMap, op::typeof(_extrema_rf), A::AbstractArray
 
     # but NaNs and missing need to be avoided as initial values
     if v0[1] isa Number && isnan(v0[1])
+        # v0 is NaN
         v0 = oftype(v0[1], Inf), oftype(v0[2], -Inf)
     elseif isunordered(v0[1])
         # v0 is missing or a third-party unordered value
-        # TODO: Some types, like BigInt, don't support typemin/typemax.
-        # So a Matrix{Union{BigInt, Missing}} can still error here.
-        v0 = typemax(nonmissingtype(Tmin)), typemin(nonmissingtype(Tmax))
+        Tminnm = nonmissingtype(Tmin)
+        Tmaxnm = nonmissingtype(Tmax)
+        if Tminnm <: Union{BitInteger, IEEEFloat, BigFloat} &&
+            Tmaxnm <: Union{BitInteger, IEEEFloat, BigFloat}
+            v0 = (typemax(Tminnm), typemin(Tmaxnm))
+        elseif !all(isunordered, A1)
+            v0 = reverse(mapreduce(f, op, Iterators.filter(!isunordered, A1)))
+        end
     end
     # v0 may have changed type.
     Tmin = v0[1] isa T ? T : typeof(v0[1])
diff --git a/test/reducedim.jl b/test/reducedim.jl
index daa0a3fbe1f92..f4767dd7a472c 100644
--- a/test/reducedim.jl
+++ b/test/reducedim.jl
@@ -608,7 +608,7 @@ end
 end
 @testset "NaN/missing test for extrema with dims #43599" begin
     for sz = (3, 10, 100)
-        for T in (Int, Float64, BigFloat)
+        for T in (Int, Float64, BigFloat, BigInt)
             Aₘ = Matrix{Union{T, Missing}}(rand(-sz:sz, sz, sz))
             Aₘ[rand(1:sz*sz, sz)] .= missing
             unordered_test_for_extrema(Aₘ)
@@ -622,9 +622,16 @@ end
         end
     end
 end
-@test_broken minimum([missing;BigInt(1)], dims = 1)
-@test_broken maximum([missing;BigInt(1)], dims = 1)
-@test_broken extrema([missing;BigInt(1)], dims = 1)
+
+@testset "minimum/maximum over dims with missing (#35308)" begin
+    for T in (Int, Float64, BigInt, BigFloat)
+        x = Union{T, Missing}[1 missing; 2 missing]
+        @test isequal(minimum(x, dims=1), reshape([1, missing], 1, :))
+        @test isequal(maximum(x, dims=1), reshape([2, missing], 1, :))
+        @test isequal(minimum(x, dims=2), reshape([missing, missing], :, 1))
+        @test isequal(maximum(x, dims=2), reshape([missing, missing], :, 1))
+    end
+end
 
 # issue #26709
 @testset "dimensional reduce with custom non-bitstype types" begin