plotting - Interpolating a plot with missing values

With

data = {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}, {7, 5}, {8, 
   4}, {10, 3}, {10, 2}, {11, 1}}

and

ListLinePlot[data, Filling -> Axis, GridLines -> Automatic, 
 Joined -> True, Mesh -> Full, PlotMarkers -> Automatic, 
 InterpolationOrder -> 2]

we get:

In reality, my datasets oftentimes have missing values. Setting f.e.

data[[3]] = {3, Missing[]}

I obtain with Off[ListLinePlot::ioproc] this picture,

nicely not-showing the missing point.

Unfortunately though, the smoothing vanished.

Could you propose a general way (plots may vary in size and may have multiple sets) to interpolate over missing values?

Thanks

EDIT

Based on the ideas presented here I have written this little function:

Smoother[v_] := If[smooth != 2, v, Select[#, #[[2]] != "na" &] & /@ v]

Turning Smoother on or off (via a button) I now get what I hoped for:

Answer

Don't know how "general" you want it, but:

data = {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6},
   {7, 5}, {8, 4}, {10, 3}, {10, 2}, {11, 1}};
data1 = data;
data1[[3]] = {3, Missing[]};

GraphicsRow[ListLinePlot[#, Filling -> Axis, GridLines -> Automatic, 
   Joined -> True, Mesh -> Full, PlotMarkers -> Automatic, 
   InterpolationOrder -> 2] & /@ {data, Select[data1, #[[2]] =!= Missing[] &]}]