What is the origin of the word 'dashboard' - as in 'car's dashboard'?

The word certainly predates the motor car, and horse-drawn carts and carriages had dashboards which were:

A board or leathern apron in the front of a vehicle, to prevent mud from being splashed by the heels of the horses upon the interior of the vehicle. Also, movable sides to a cart for the same purpose (Halliwell). (OED)

Doubtless this is where the name comes from. But why were they called dashboards? The verb dash means to strike or hit. Was it that the earliest dashboard was just a hit board for mud, stones etc?

It appears that you are right on your assumption: World Wide Words has the story:

• The sense of dash is the one that refers to the “violent throwing and breaking of water or other liquids upon or against anything”, as the Oxford English Dictionary puts it.

• The dashboard was a wooden board, or a leather apron like the one that the article mentions. It was placed at the front of a carriage, sleigh or other vehicle to catch the mud or water thrown up by the horses’ hooves and stop it from soaking the driver and his passengers. We would now call it a kind of mudguard.

• The very earliest examples of the original sense of dashboard are from the second decade of the nineteenth century. This is a slightly later appearance:

  • On Monday evening, Lord Lyndhurst was driving a gig near Guildford, when the horse began to kick and plunge, and at length breaking the dash-board, his Lordship and his friend jumped out, and sustained no injuries. The Morning Post (London), 24 July 1832.

• Since horse-drawn vehicles were hardly new, it had presumably had other names before this, although the only other one I can turn up is splash-board, which is contemporary with it.

• Early motor vehicles left the driver totally exposed to the weather, so the dashboard wasn’t as useless as suggested by the anonymous writer I’ve quoted. It did protect the legs of the driver against wind and rain. A similar design appeared on trolley cars, trams and other vehicles (such as the one below) and had the same name.

• As vehicle designs evolved, a windscreen (windshield) was put in place above the dashboard and the latter became a handy place to put the instruments. But it kept its name.
• It has recently stepped even further away from its origins by being borrowed for a computer display that shows useful real-time data such as the time, weather, news headlines, stock prices and phone numbers. It’s a long way from horse-drawn carriages on muddy roads.

Dashboard ancient usage:

• While the term “dashboard” didn’t work its way into popular English until the 1800’s, the concept of a “dashboard” existed long before then. In fact, Mesopotamian chariots dating as far back as 3,000 BC employed similar guards against mud and rocks.

(thehogring.com)