Your data is a bit noisy which is normal for experimental data:

yourData = Import["~/tmp/ASmappa-niccolini90-0.dat", "TSV"];

you will want to use your path to your data.

dataForInterpolate = yourData /. {x_, y_, z_} :> {{x, y}, z};

zInterpolation = Interpolation@dataForInterpolate

Interpolate complains that your data is not on a regular (i.e., rectangular grid) so it reduces the order of interpolation.

Plot3D[zInterpolation[x, y], {x, 0.005, 5}, {y, 0, 3.95`}]

That means that your first-order derivatives will be extra noisy. However, we can proceed:

dzdxInterpolation = Derivative[1, 0][zInterpolation]
dzdyInterpolation = Derivative[0, 1][zInterpolation]

NIntegrate[
 Sqrt[1 + dzdxInterpolation[x, y]^2 + dzdyInterpolation[x, y]^2], {x, 
  0.005, 5}, {y, 0, , 3.95`}]

Complains that convergences is slow. This is likely from your derivatives. Integrate prints a result and an error estimate.

Possible fixes are to resample your data onto a regular grid, smooth a bit, and use Splines to approximate your data.

I am so grateful for your time and help. I applied your method to my calculations, but there is an error! I will attach the link for downloading my full data. Can you please check the example data in this case? Thank you so much in advance

exampleData = Flatten[Table[{{x, y}, f {x, y}}, {x, 0, 1, .01}, {y, 0, 1, .01}], 1] 

https://drive.google.com/file/d/1t0ZWf6BdzatNFC9DIr4yqulEz6c2fCe8/view?usp=sharing

Use the following code to construct $f$ from you data.

f[data_] := Interpolation(data, Method -> "RBF");

If you post your data together with $A$ and $A_R$ (i.e. the domain of integration), then I can show you a complete code on how to evaluate your formula numerically using NIntegrate.

Can I kindly ask you what is the code for RBF? I can't find something good to transform z to f(x,y).