Ulysses S. Grant was arrested for "fast driving" at least once, in April of 1866, which is after the U.S. Civil War ended but before Grant was elected President.
This was reported in the Alexandria (VA) Gazette on April 12, 1866 (but copied from another newspaper, so could be earlier):
GENERAL GRANT'S CASE -- This morning,
the Sergeant of the Second Precinct reports
the case of Gen. U. S. Grant, arrested for fast
driving, as settled by the General paying the
fine. We are informed that when the policeman
went to Army Head Quarters and laid his
warrant on the table before the General, that
officer looked at it, and turning to the servant
of the law, remarked, "I suppose you take a pride
in this?" The officer said: -- "No, General I
only do it because it is my duty." The General
pleasantly intimated that under martial law the
tables might be turned, and the server of the
warrant sent to the guard-house. The officer
replied that the General might do as he pleased
but he had performed his duty, adding, "I
also did my duty under you, General, at Vicksburg,
and you did not find fault with me then."
The General immediately acknowledged the
service of the warrant, appeared before Justice
Walter, and paid the fine. -- Wash. Star.
And in the Camden (NJ) Weekly Journal on April 20, 1866 (again copied from another newspaper):
General Grant Arrested for Fast Driving.
On Saturday, while General Grant
was exercising his fast gray nag on
Fourteenth-street, officers Bailey and
Crown, after a sharp race, arrested him
for fast driving. General Grant offered
to pay the usual fine imposed in such
cases, which, of course, they could not
receive; but the General expressed his
doubts of their authority to arrest him,
and drove off. The case was duly reported to Superintendent Richards.
It is stated that this street is becoming
a common racing ground, and that a
large number of arrests for violations
of the ordinance prohibiting fast driving
are made every pleasant day, when
those who delight in "speed" are out
exercising their "stock."
National Intelligencer, 9th.
Since the second one says "9th" and refers to "Saturday," it's possible that the incident occurred on Saturday, April 7, 1866.
This may not have been the only time that Grant was stopped for speeding. An 1886 story, written in the year after Grant's death, relates "how he was twice arrested for fast driving" (Opelousas Courier, January 9, 1886).
The September 19, 1867, Indianapolis (IN) Daily Herald contains a short news item about Grant running over a child and repeats the statement that Grant had been arrested more than once:
Fast Horse Grant.
"General Grant, who has been once or twice
arrested for fast driving, is reported by the
police to-day as having run over a small boy,
yesterday evening, while riding out and
driving very rapidly. He stopped to see the
boy properly cared for, and ordered all bills
sent to him for payment."
It must be a great consolation to the father
ot the "small boy" to know that the little fellow
was run over by a great General, one who
never surrenders, and that he grandly "ordered
all bills sent to him." Some of these
fine days some unappreciative boor may vulgarly
take the laws into his own hands and
send Uncle Sam Grant home with a battered
head. How shocking to have the brains of a
great General imperilled by a brickbat in the
hands of a stout hod carrier. -- New Albany Commercial
(Chronicling America does not identify any holdings for the Alexandria Gazette or the Washington Star for the relevant dates.)