The thought occurred to me that, after obtaining the values by formulae, you
should convert them to values so that they are no longer linked. Suggested is
this:

Sub GetValsByFormula()
Dim FNs As Variant
Dim FN As String, Pth As String
Dim i As Integer
Dim r As Range

On Error GoTo ExitProc
FNs = Application.GetOpenFilename _
("Excel Files(*.xls), *.xls", MultiSelect:=True)
If TypeName(FNs) = "Boolean" Then Exit Sub
FN = Dir(FNs(LBound(FNs)))
Pth = Left(FNs(LBound(FNs)), Len(FNs(LBound(FNs))) - Len(FN))
For i = 1 To UBound(FNs)
FN = Dir(FNs(i))
Cells(i, 1).Formula = "= '" & Pth & "[" & FN & "]Sheet1'!A1"
Cells(i, 2).Formula = "= '" & Pth & "[" & FN & "]Sheet1'!B1"
Next
Set r = Range(Cells(1, 1), Cells(UBound(FNs), 2))
r.Copy
r.PasteSpecial (xlPasteValues)
Application.CutCopyMode = False
ExitProc:
End Sub

Regards,
Greg