My whippet has just turned 1 and is in good health, generally.

However, she started kind of retching and coughing a few days ago. It wasn't too bad initially but then, on Thursday, she had quite a coughing fit and my other half took her to the vet.

As I wasn't there at the vet's I'm a bit disappointed that we haven't clarified whether or not she has kennel cough. Other half reported that the vet said she had a cough, gave her an anti-inflammatory injection, sent us home and advised to give her small doses of benylin, which we have done. Whether it is kennel cough or not we do not know but the uncertainty is causing us problems as I can't take her to her usual dog sitters', which I use when I am working and my other half is away, so I am having to work from home with her.

I thought she was better yesterday, and she shows no signs of ill health and still races around everywhere. However, this morning she wheezed again and still has the odd coughing fit. There is no phlegm or runny nose. She also does not have diarrhoea or vomiting.

I'm not sure what to do next. The vet said we should take her back in on Monday or Tuesday if she is still coughing.

Does anyone have any experience of kennel cough or general coughs and any advice as to what this might be?

It sounds like Kennel Cough to me. It has a really distinct hoarse wretchy cough like a hairball or something stuck at the back of the throat. I do the same as your vet has done. Benylin cough syrup (which the dogs seem to love) and a jab (obviously given by vet not myself) of anti inflams to ease the throat and trachea. Then just plenty of love and care (and on isolation from other dogs eg NO walking at anytime, anywhere) The bug can live on grass for a while until it finds another host. There is a big problem somewhere near Thatcham where the southern counties showground is.

There should be a 21 day period of isolation after the last signs of the cough as this is how long the virus can survive for.

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SOMETIMES YOUR KNIGHT IN SHINING ARMOUR IS MERELY AN IDIOT WRAPPED IN TINFOIL

Sounds like classic kennel cough symptoms.
You will be working from home for a while yet

It does sound like kennel cough. However, heart disease is fairly common in whippets, so please do bring her back to the vet's if she's not improved by Monday or Tuesday as coughing can also be a sign of heart disease.

Also, hopefully your sitter has been able to inform the other clients that kennel cough might be going through her service.