Of American-English origin, the colloquial phrase to go (and) jump in(to) the lake means: to go away and stop being a nuisance. It is chiefly used in the imperative as a contemptuous dismissal. The image is probably of somebody jumping into a lake and drowning.

This phrase occurs, for example, with punning reference to the U.S. politician and former television news anchor Kari Lake (born 1969) in Don’t nominate Lake for Arizona governor, by the Washington Post columnist Henry Olsen, published in the Messenger-Inquirer (Owensboro, Kentucky, USA) of Thursday 7^th July 2022:

Early voting for Arizona’s primary elections started on Tuesday. Republicans can do something good for their country and party by defeating Kari Lake.
Lake, once an anchor on Phoenix’s Fox News outlet, is former president Donald Trump’s choice for governor. […]
[…]
Arizona Republicans can’t take that risk. They should instead give Kari a clear message: Go jump in the lake.

The earliest occurrences of the phrase to go (and) jump in(to) the lake that I have found are as follows, in chronological order:

1-: From one of the miscellaneous comments on baseball making up Diamond Drift, published in The Boston Daily Globe (Boston, Massachusetts, USA) of Sunday 12^th August 1883:

Decker ought to write a book on what he knows about umpiring—it would only take him a second—and then go and jump into the lake.—[Cleveland Leader.

2-: From the title given to the following letter, published in The Daily Inter Ocean (Chicago, Illinois, USA) of Thursday 12^th July 1888:

GO JUMP IN THE LAKE.

Chicago, Ill., July 11.—To the Sporting Editor.—Dear Sir—being a stranger and in a strange city I thaught that I would write you for a little advise I have a system for beating the races whitch I have done for the last two seasons but owing to curcamstances over whitch I have no controle I am about broke. The advise whitch I want is how would I be able to get a man with $300 to start the System along the only explination I can give you now is this. a man can start with $300 and in 10 days it will duble to $600 twenty days $1,200 and so on doubling itself every 10 days. now you will say that this is all bosh and that I am crazy, but I dont ask any one to take my word for it I can produce solid facts that will convince any sane man that if he had started with $300 at New Orleans or Memphis and followed it up to the present time he would have made over $30,000.00 I dont want any one to give me cash money for my system neither do I want half of all they make off it. I will put anybody on that means business and take the profits for the first 10 days and the man can handle his own money possiablly you will think that if I have such a good thing I ought to have no trouble in getting a man but I guess you are not in my position now what would you advise me to do
Hoping to hear from you soon through the Ocean I remain Yours Truely
A READER.

3-: From The Racine Daily Journal (Racine, Wisconsin, USA) of Tuesday 3^rd June 1890:

TIT FOR TAT.
How a Young Lady Got Even With Her Intended Husband.

Gossips on the North Side are much agitated over a marriage that was to have taken place on two different occasions, but did not. The first time the young lady was all ready, and had on the bridal robe, when the young man telegraphed from Milwaukee that he could not come and the event must be declared off. On Sunday last the marriage was again announced to come off. The young man, with his father and mother were promptly on hand, but the intended bride had mysteriously disappeared and not the slightest trace of her could be found. Those interested were on the verge of notifying the police of the girls’ [sic] disappearance, when a letter was received from Chicago. It was from the young lady and informed the young man to “go and jump into the lake; she would never marry him; that he played a dirty trick on her once and now it was her turn.” The young man packed his grip and silently stole away.

4-: From the St. Louis Post-Dispatch (St. Louis, Missouri, USA) of Monday 5^th November 1894:

They Felt Better.

From the Chicago Record.
This man was one of the cheap creatures who loaf in front of the hotels and cigar stores and look at women. He thought the girl would simply blush and hurry on as the others had done when he said: “Hello, sis.” But she didn’t. She turned on him.
“You’re a nice man, ain’t you?” said she. “Do you think a woman with any sense would look at a big loafer like you? I’ll bet you haven’t done an honest day’s work in a year. Say, it ain’t a bit smart to stand out here and insult women when they’re goin’ by. If I was as low down as you I’d go jump in the lake.”
“Give it to him,” said a man who bad stopped to listen.
The girl noticed the crowd gathering and hurried on. The big man called the cheap “masher” a dreadful name and the latter went inside. But everyone who happened to be there felt a little better because of the incident and forgave the big man for what he said.

A synonymous phrase, to go and jump into Wynyard lake, occurs in the following from the North-Eastern Daily Gazette (Middlesbrough, Yorkshire, England) of Thursday 15^th August 1889—this phrase probably refers to the artificial lake at Wynyard Park, the principal residence of the Marquesses of Londonderry, near Stockton-on-Tees, County Durham, England:

THE HOME RULE VAN AT NORTON.

The literature van of the Home Rule Union left West Hartlepool yesterday at noon, and, travelling by Sedgefield, reached Norton a little before seven o’clock. Half an hour later a public meeting was held on the Green, which was attended by some 600 persons, in which there was a fair sprinkling of Conservatives.—Mr R. H. Holliday presided, and in opening the proceedings he said they had met to discuss a very important question, and he bespoke for the speakers a fair and impartial hearing. It was a question which had occupied the attention of the public for some time, and would continue to engross their attention till it was settled one way or another. The Tories told them that there was no need for change in Ireland. (A Voice: “Oh, dear no.” Another Voice: “Go and jump into Wynyard lake.”)