My understanding is that if you have a proper international carnet then there is no time restriction on leaving a vehicle in Senegal, as long as your carnet is still in date, but I am willing to be corrected on that one.
When I travelled down to The Gambia I crossed into Senegal via Rosso (not particularly recommended!), and even though I had a carnet the customs official put vehicle details in my passport. When I travelled back from the Gambia I crossed via Diama Dam and nothing was written in my passport concerning the bike! Go figure!
With regards a secure place to leave your car, you might (and this is only a suggestion) be able to come to some arrangement with a place called Zebrabar near St Louis. This haven of tranquility is a campsite/collection of bungalows just south of St Louis. It is run by a Swiss couple and is a very popular stopping off point for travellers on their way through Senegal. Its clean, has a fantastic beach, hot showers, good food and most importantly the  is cold!!
It is not too far from Dakar, you could certainly get a taxi to the airport. Try them. Their website is http://www.flueckiger.net/zebrabar/ Even if they cannot help you they may be able to point you in the direction of someone who can.
Beware though, most people go there for one night and find that they end up staying a few days. It's difficult to bring yourself to leave the place!!
Enjoy!
|