I worked at a restaurant where we had a cake with this icing and it was divine and kept for several days, refrigerated or course.
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Try stabilizing it with gelatin. Whip the cream, sugar, and any flavorings to soft peaks. Bloom 1 sheet gelatin per 2 cups cream (or maybe even use more gelatin), squeeze the excess water out of the bloomed gelatin, melt the gelatin in a saucepan with a tablespoon of water, and slowly pour the melted gelatin into the cream with the mixer running. Whip to firm peaks.
In addition to the above I would suggest beating it in a processor, not a mixer. This willIncorporate slightly less air and therefore make it more stable. A trick I learned in the cake bible
Rose's Cake Bible also has instructions for stabilizing with cornstarch. For two cups of whipped cream: boil 1/4 cup heavy cream, 2 tbsp powdered sugar, and 1 tsp cornstarch, stirring constantly, for just a few seconds to thicken the cream. Stir in vanilla or other flavoring, then chill. Whip 3/4 cup cold heavy cream to soft peaks, then add the starch mixture in a steady stream and beat to stiff peaks. Rose says this cream will not "water out" for 24 hours.
I've also had luck with the Cook's County strawberry cream cake, which stabilizes the cream with cream cheese. The flavor is different, but very good. The cake is on pg. 93 of "Blue Ribbon Desserts."