It sounds like a 1D array of 2D arrays would be the best approach.
map or broadcast with eachslice can yield a 1D array of 2D arrays:
julia> x = rand(3,3,4);
julia> inv.(eachslice(x, dims=3))
4-element Array{Array{Float64,2},1}:
[-1.6590312226570239 2.0793019606694854 0.5372029679273337; -0.41725584494784745 -0.16227671804840568 1.4356241190356016; 3.1581583049406534 -1.8836845469856938 -1.3709070403858319]
[2.2504272336124984 -1.3387872100956475 -1.3175125593090484; -0.6955200736976385 2.3904260659293395 -0.15231503786728723; -1.6588907341843613 -5.504678105886818 6.354210163417449]
[-6.522265014454815 11.007596635562658 -0.8997442890195426; 5.918653872368978 -6.235209223176827 -0.20055712370467133; 3.1672517888989637 -7.134447496548201 2.0670158780550607]
[1.257897188987869 0.5534734488273791 -3.7933556798265524; -1.3639997740291203 1.2156395483916482 3.2530995922802717; -1.1021239862109267 -3.2404755836203387 21.985361866828296]
julia> map(inv, eachslice(x, dims=3))
4-element Array{Array{Float64,2},1}:
[-1.6590312226570239 2.0793019606694854 0.5372029679273337; -0.41725584494784745 -0.16227671804840568 1.4356241190356016; 3.1581583049406534 -1.8836845469856938 -1.3709070403858319]
[2.2504272336124984 -1.3387872100956475 -1.3175125593090484; -0.6955200736976385 2.3904260659293395 -0.15231503786728723; -1.6588907341843613 -5.504678105886818 6.354210163417449]
[-6.522265014454815 11.007596635562658 -0.8997442890195426; 5.918653872368978 -6.235209223176827 -0.20055712370467133; 3.1672517888989637 -7.134447496548201 2.0670158780550607]
[1.257897188987869 0.5534734488273791 -3.7933556798265524; -1.3639997740291203 1.2156395483916482 3.2530995922802717; -1.1021239862109267 -3.2404755836203387 21.985361866828296]
julia> mapslices(inv, x, dims=(1,2))
3×3×4 Array{Float64,3}:
[:, :, 1] =
-1.65903 2.0793 0.537203
-0.417256 -0.162277 1.43562
3.15816 -1.88368 -1.37091
[:, :, 2] =
2.25043 -1.33879 -1.31751
-0.69552 2.39043 -0.152315
-1.65889 -5.50468 6.35421
[:, :, 3] =
-6.52227 11.0076 -0.899744
5.91865 -6.23521 -0.200557
3.16725 -7.13445 2.06702
[:, :, 4] =
1.2579 0.553473 -3.79336
-1.364 1.21564 3.2531
-1.10212 -3.24048 21.9854