A first steam boat

By the time that the Sankey Navigation (canal from the Mersey to St Helens - the first such in Britain) was opened in 1757, colliery owners in the region of St Helens were using Newcomen steam engines to pump water from their collieries. The Newcomen engine was very suitable for pumps, since the motion provided was just up and down. Converting this into rotary motion for paddles in a boat would need cranks and shafts. It was also inefficient - though collieries had ample low grade coal to spare - and Watt later introduced (patented) improvements.
Most boats and ships were made of wood at that date. Some canal boats, however, were of iron - since it allowed a much shallower draft for the same load. Iron was especially suited to the introduction of a steam engine since it was not liable to burn.
A record by colliery owner, William Bromilow, of an early attempt to combine steam power with a hull (possibly of iron) is presented below. Since the vessel is reported to have travelled from the Sankey navigation to Runcorn where it entered the Bridgewater canal to Manchester, it had to cross the tidal Mersey - so, arguably, involving a sea voyage. There were similar developments in Scotland and America about this date, so the claim to be the first needs to be treated as indicative rather than definitive.

[from Liverpool Mercury - Friday 20 July 1832]:
TO THE EDITOR. SIR, - Seeing in your paper of the 30th ult., an inquiry relative to the first inventor of steam-boats; also some inquiries relative to one constructed at St. Helen's, by a John Smith, and having seen no answer I willingly give such information as I am in possession of, and shall have pleasure if it leads to any thing like a remuneration to the family, some of whom are still living in St. Helen's, and are only in very moderate circumstances.
The engine in the boat alluded to, and which is generally supposed to be the first invented, was constructed for propelling boats by steam, as before stated, by Smith of St. Helens, in the year 1793[sic, probably later], and her first excursion was down the Sankey to Newton Races, in June the same year, laden with passengers. On the Saturday following, she sailed to Runcorn, from thence down the Duke of Bridgewater's Canal to Manchester. On her arrival there, such was the curiosity at this wonderful (and as some would have it) this mad idea, that thousands of the people came from all directions to see what their eyes could not believe, nor their senses understand; and, indeed, such were the numbers, and such the curiosity that this vessel excited, that Smith was obliged, for the safety of his property, to give notice that no one would be allowed to come on board of her, excepting those who paid a certain sum. This exasperated the populace to such an extent, that a party of mechanics immediately got possession of, and almost destroyed her. Amongst the visitors was Mr. Sherratt, of the firm of Bateman and Sherratt, of Manchester; also several other respectable engineers of the same place, for whom it is unnecessary to name.
So far as memory serves me, (after a lapse of 39 years,) the following is a short description of this wonderful discovery; but having made no memorandums of the circumstance at the time, and I may say, being then young, and to a certain extent, like the rest of my friends, incredulous, I never anticipated what is almost to every one in the present day so common. The vessel had on her an engine on the old atmospheric principle, was worked with a beam, connecting-rod, double-crank in a horizontal line, and with seven paddles on each side, which propelled her after the rate of about two miles an hour. John Smith was a rude, self-taught mechanic, and was supported by a Thomas Baldwin, at that time of St. Helens, and [Baldwin] was the first aeronaut who ever ascended in a balloon, either in this or the adjoining counties. Perhaps, I may observe, that the vessel or boat was purchased at Liverpool, and on Smith's informing the parties from whom he bought it, what his intentions were, he was treated as an insane person; he was laughed at by one, insulted by another, and pitied generally; but, having money with him, he was allowed to purchase her. On being questioned and laughed at by a merchant at the time the purchase was made, he replied, "those may laugh who will, but my opinion is, before twenty years are over, you will see this river [Mersey] covered with smoke".
William Bromilow, Morton Bank, near St Helens

[from Billinge's Advertiser of 26 June 1797]: An unusual occurrence took place at Newton Common, on Friday the 16th inst.: being the last day of the races there, a vessel, heavily laden with copper slag, passed along the Sankey Canal, without the aid of hawlers [sic, haulers] or rowers; the oars performing 18 strokes a minute, by the application of steam only. On enquiry since made, it appears that the vessel, after a course of ten miles, returned the same evening to St. Helen's, whence it had set out. The form and motion of the oars is not easily described, but it bids fair to be ranked among the most useful of modern inventions

Possibly motivated by John Smith's pioneering boat, soon afterwards, in 1799, the Duke of Bridgewater had been involved with a steam tug which ran along the Bridgewater canal in Worsley. Fulton [later to develop American steam-boats] suggested to the Duke of Bridgewater the feasibility of towing boats on his canal (completed 1776) by steamboat. The Duke approved a trial and a boat was constructed: the engine and gearing were constructed by Salford firm Sherratt and Bateman, with the hull being made in Worsley Yard (now Worsley Green). It is also known that Capt. Shanks of the Royal Naval Dockyard at Deptford played a part. The boat [facetiously named Bonaparte since unpopular] was a stern-wheeler, most probably of iron, and was tested in trials towing barges but did not lead to further use, since it was slow and was thought to damage the canal banks. The engine was removed and used for other purposes.
  Plan and Section of a steam boat and engine for his Grace the Duke of Bridgewater (1799) signed William Sherratt: [from Institute of Mechanical Engineers]

Iron steam vessel 1815. Brief reports - of which no more detail is known to me.

[from Liverpool Mercury - Friday 19 May 1815]:
STEAM BOAT ON THE MERSEY. We understand an Iron Boat is now constructing for our River, to be navigated by steam; it is intended to ply between Liverpool and Runcorn.

[from Manchester Mercury - Tuesday 30 May 1815]:
Steam Boat on the Mersey. We understand an Iron Boat is now constructing for that River, to be navigated by steam; it is intended, we are told, to ply between Liverpool and Runcorn, for the conveyance of stone to be used in the new docks at Liverpool.